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INSTITUTES 


OF 


NATURAL PHILOSOPHY 


THEORETICAL AND PRACTICAL. 


BY WILLIAM ENFIELD, LL. D. 


2 73 
3o 


\ . 


WITH SOME CORRECTIONS ; 

CHANGE IN THE ORDER OF THE BRANCHES', 

AND THE ADDITION OF 

AN APPENDIX TO THE ASTRONOMICAL PART, 

' S 

SELECTED FROM 

MR. EWING’S PRACTICAL ASTRONOMY. 


BY SAMUEL WEBBER, A. M. A. A. S. 

Late President of Harvard College. 


FOURTH AMERICAN EDITION, WITH IMPROVEMENTS. 

Ornnis philosophise difficultas in eo versari videtur, ut a phsenomenis motuum investigemus vires naturae, deinde 
ab his viribus demonstremus phenomena reliqua.— Newton. 


BOSTON: 

PUBLISHED BY CUMMINGS, HILLIARD, & CO. AT THE BOSTON BOOKSTORE, 

NO. 1 CORNHILL. 

University Press.Hilliard 4" Metcalf. 

1824. 











QCai 
. £ 56 
i<2Zf 


DISTRICT OF MASSACHUSETTS, to 7 vit: 

District Clerk's Office. 

BE IT REMEMBERED, that on the twenty third day of December A. D. 1819, in the forty-fourth year of the Independence of the United States 
of America, Cummings & Hilliard, of the said district, have deposited in this office the title of a book, the right whereof they claim as proprietors, in 
the words following, to wit: 

“ Institutes of Natural Philosophy, theoretical and practical—By William Enfield, LL. D. With some corrections ; change in the order of the 
branches ; and the addition of an Appendix to the astronomical part, selected from Mr. Ewing’s Practical Astronomy. By Samuel Webber, late 
President of Harvard College. Third American edition, with improvements. Omnis philosophise difficultas in eo versari videtur, ut a phaenomenis 
inotuum investigemus vires nature, deinde ab his viribus demonstremus phenomena reliqua— Newton” 

In conlormity to the act of the Congress of the United States, entitled “ An act for the encouragement of learning, by securing the copies of maps, 
chart?, and books, to the authors and proprietors of such copies, during the times therein mentioned and also to an act, entitled “ An act supplemen¬ 
tary to an act, entitled * An act for the encouragement of learning, by securing the copies of maps, charts, and books, to the authors and pro- 

S rietors of such copies, during the times therein mentioned and extending the benefits thereof to the arts of designing, engraving, and etching 
istorieal and other prints.’ 5 JOHN W. DAVIS, 

Clerk of the District of Massachusetts. 



TO THE 

REV. JOSEPH PRIESTLEY, 

DOCTOR OF LAWS, FELLOW OF THE ROYAL SOCIETY, &C, 

IN TESTIMONY OF RESPECT 

FOR A CHARACTER EMINENTLY DISTINGUISHED 

BY 

COMPREHENSIVE AND ENLARGED VIEWS OF SCIENCE, 
ASSIDUOUS AND SUCCESSFUL RESEARCHES INTO NATURE, 

AN ARDENT LOVE OF TRUTH, 

INDEFATIGABLE ZEAL IN THE SERVICE OF RELIGION, 
SIMPLICITY OF MANNERS, 

And 

AN ACTIVE SPIRIT OF PHILANTHROPY, 

THIS WORK 

IS INSCRIBED 

BY HIS AFFECTIONATE FRIEND, 

AND OBEDIENT SERVANT, 


WILLIAM ENFIELD. 


> 




PREFACE. 


Nothing can be an adequate apology for obtruding upon the world a 
new Elementary Work, in a branch of Science already well understood, 
except the plea of utility. It is wholly upon this ground, that I venture to 
submit the following Treatise to the public inspection. 

The difficulty which I met with, in providing my Classes* with a Text¬ 
book in Natural Philosophy, neither, on the one hand, materially deficient 
in Mathematical Demonstration, nor, on the other, too copious, or too 
abstruse, for the purpose of elementary instruction, first suggested the idea 
of this w ork. And the apprehension that others may have met with the 
same difficulty, induces me to make it public, in hopes that it may be of 
some use to those who wish to study, or to teach, this science systemati¬ 
cally. 

To that class of readers who are satisfied with general views, this work 
will be of little service. Sketches of philosophy, sufficiently comprehensive 
to answer their purpose, will easily be found. But the knowledge, which 
is gathered up in this cursory manner, must unavoidably be superficial, and 
will in many particulars, be confused and inaccurate. What Cicero says of 
philosophy in general, is particularly true of natural Philosophy: Difficile 
est enirn in philosophid pauca esse ei nota , cui non sint nut plevaque , ant om¬ 
nia .f It may be laid down as an universal maxim, that there is no easy 
method of obtaining excellence. The small portion of learning, or science, 
which is to be acquired by the help of facilitating expedients, has been justly 
compared to a temporary edifice built for a day.f It is as unreasonable to 
Jiope to acquire knowledge without undergoing the labour by which it is 
usually gained, as it would be to expect that an acorn will become an oak, 
without passing through the ordinary process of vegetation. 


♦ In the Warrington Academy. 


t Tuscul. Quaest. II. 1. 


$ Knox on Liberal Education. § 9. 


Vi 


PREFACE. 


All the knowledge of Natural Philosophy which can be acquired by 
cursory reading, without the assistance of mathematical learning, must con¬ 
sist in an acquaintance with leading facts and- general conclusions. To 
understand the manner in which the laws of nature have been inferred from 
these facts, and to be able with certainty and precision to apply these laws 
to the explanation of particular phenomena, necessarily requires a previous 
knowledge of the elements of Geometry, Trigonometry, the Conic Sections, 
and Algebra. A mechanic, who should set about making a machine with¬ 
out the requisite tools, would not act more absurdly, than a student who 
should attempt to understand the science of Natural Philosophy without 
these helps. A preceptor, who professes to teach this science in the easy 
and amusing method of experiment alone, is an architect without his rule, 
plumb-line, and compasses. 

Facts are, it is true, the materials of science ; and much praise is un¬ 
questionably due to those who have increased the public store, by new 
experiments accurately made, and faithfully related. But it is not in the 
mere knowledge, nor even in the discovery of facts, that philosophy con¬ 
sists. One who proceeds thus far, is an experimentalist; but he alone, 
who, by examining the nature, and observing the relation of facts, arrives at 
general truths, is a philosopher. A moderate share of industry may suffice 
for the former : patient attention, deep reflection, and acute penetration, 
are necessary in the latter. It is therefore no wonder, that amongst many 
experimentalists there should be few philosophers. 

The hardy perseverance, and the vigorous exertions, which are neces¬ 
sary to form this character, are so contrary to that effeminacy and frivolity 
which distinguish the present age, that, if it were not for the provision 
which is made in our universities, and other seminaries, for the propagation 
of sound learning of every kind, there would be some reason to apprehend, 
that all the more abstruse and difficult branches of science would be exclud¬ 
ed from the modern system of education, and consequently would fall into 
disesteem and neglect. 

It is by no means the intention of this treatise to encourage the indolent 
spirit of the times, by opening a bye-path to the Temple of Philosophy. 
The known and beaten road is the safest and the best. It has been with a 
view of assisting the student in his progress, that I have attempted to ar- 


r 


PREFACE. vii 

range the leading truths of Natural Philosophy in a perspicuous method, 
and to demonstrate them with conciseness ; adding a brief description of 
experiments, adapted to illustrate and confirm the propositions to which 
they are respectively subjoined. 

Being more desirous to be useful than to appear original, I have freely 
selected from a variety of authors such materials as suited my design. 
Those who are conversant with this class of writers will perceive that, 
amongst many others, I have made use of the works of Newton, Keil, 
Whiston , Gravesande , Cotes , Smith , Helsham , Rowning , and lastly, Rutlier- 
forth , whose arrangement I have in part adopted. 

With respect to any inaccuracies or mistakes which may have escaped 
my attention, I must rely on that candour, which those who are best ac¬ 
quainted with the extent and difficulty of this undertaking will be most 
inclined to exercise. 


i 


ADVERTISEMENT 

TO THE SECOND LONDON EDITION, 

BY THE EDITOR. 


In laying before the public a new edition of “ The Institutes of Natural 
Philosophy , by the late Dr. Enfield,’’ the Editor feels it incumbent on him 
to assure the reader, that he has endeavoured, as far as was consistent with 
an elementary book, to avail himself of those advantages which the publica¬ 
tion of new discoveries, and new works in science, has afforded him ; and 
although the limits of an advertisement will not allow him to particularize 
all the additions that will be found interwoven with the various parts of the 
volume, yet it may be expected that, in this place, some notice should be 
taken of the most material of them ; and it is presumed that the following 
account will be deemed sufficient for the purpose. 

In the first book, the propositions on the divisibility of matter, and the 
attraction of cohesion, are more fully discussed, and a very useful corollary is 
drawn from that on the attraction of capillary tubes. To the first and third 
propositions of the second book, considerable additions are subjoined ; and in 
the second chapter is inserted a new proposition, from which, in conjunction 
with others, are deduced many corollaries and scholiums, connected with the 
remaining parts of the book.* Several examples are also given in the two 
first sections of the fifth chapter, which will be found useful to the young 
student, as illustrative of the theory of falling bodies. 

In the third book is given, independently of the additions noticed in the 
margin,! an important proposition on the specific gravities of bodies, with 
which are connected examples, and a table of the comparative weights of 
many of the most useful subtances in nature. Descriptions, accompanied 

* See Prop. A, (p. 9.) and Cors. and Schols. kc. to Prop. 14,17,24, 26, 28,30, 31,36, 44, 46, 49, 52, 53, 54, 57, 
and 58. 

t See additions to Prop. 3, 6, IB, 13,18, A, (p. 60,) 50, and 55. 


ADVERTISEMENT. 


IX 


with figures, are likewise given of the Pyrometer, Air-Pump, Barometer, 
with its application to the measuring of altitudes, &jc. Fahrenheit’s Ther¬ 
mometer, with a Table of heat: different kinds of Hygrometers, the Steam- 
Engine, and the Hydrometer. 

The principal additions to the book of Optics will be found connected 
with the propositions mentioned below ;* in the course of which are intro¬ 
duced Mr. Delaval’s Theory of Colours ; brief accounts of Hr. Blair’s 
Achromatic Lenses, and Dr. Herschel’s grand Telescope. 

On the subject of Astronomy, are arranged under the different articles 
several useful Tables, and the important discoveries of the illustrious I)r. 
Herschel, which have been carefully selected from the last twenty volumes 
of the Philosophical Transactions. The reference in the margin,f will 
direct the reader to those propositions to which the most material additions 
are subjoined. 

Some valuable treatises, on Magnetism and Electricity, particularly those 
of Mr. Cavallo, having appeared since the original publication of this volume, 
it was thought necessary very considerably to enlarge this part of the work ; 
and it is hoped that the principal discoveries in these branches of science 
will now be found under their respective heads. 

By the suggestion of a friend, on whose judgment the public has long 
placed great confidence, it has been deemed proper that the first principles 
of chemistry should form a part of the present volume and although we 
have chiefly confined ourselves to the interesting discoveries of the philoso¬ 
phers, Black, Priestley, and Lavoisier, on Heat and the Factitious Airs, it is 
nevertheless presumed, that enough has been said on these subjects to ren¬ 
der the doctrines and introductory practice to modern chemistry perfectly 
intelligible to any person who may be desirous of farther prosecuting the 
study of this amusing and useful science. 

The reader ought to be apprized, that besides additions to the old plates, 
two new ones are now T given:—one, as already noticed, accompanying descrip- 

* See Prop 5, 13, 22, 42. B, (p. 120) 61, 62, 66, 68, 69, 76, D, (p. 133) 94, 122, 128, 144, and 145. 

t See Def. 1, 12. Prop. 8, 16, 20, 32, 35, 39, 51, 57, 72, 78, 79, 83, 109, 116, 117, 118, 119, 120, 123,136, 167,168, 
177, 179, A, B, (p. 323) 182, 183. 

| [It has been deemed best to omit the Introduction to Chemistry. As an elementary treatise, it has been found defec¬ 
tive, and as far as our information extends, it has not been generally used in those seminaries, where these Institutes are 
taught, the Chemical Professors generally recommending their respective favourite authors.] 

b 


X 


ADVERTISEMENT. 


tions of several pneumatic and hydraulic machines, and the other containing 
figures relating to subjects in magnetism, electricity, and chemistry. 

It is hoped that the augmentations to the volume, although they com¬ 
pose about one third of the whole work, w ill be found such as ought, at this 
period, to be comprehended in an elementary book of science ; and that the 
speculations of Dr. Herschel, towards the end of the astronomical part, will 
not be considered as an exception : they are at least the speculations of a 
great mind, and capable of exciting, in every well-disposed heart, emotions 
of interest and exquisite pleasure, inasmuch as they lead to the grandest and 
most sublime notions of the great Author of the universe. 

The editor will only add, that in the additions to this work, he has 
uniformly aimed at conciseness ; and he will consider his exertions well 
rewarded, if it be found, by a candid and discerning public, that he has not 
sacrificed perspicuity to brevity, and that he has not omitted, within his 
prescribed limits, any material article that might serve to render the original 
work, in its present enlarged state, generally acceptable and useful. 

May 14, I79y, 


ADVERTISEMENT 

TO THE FIRST AMERICAN EDITION. 


The principal object in undertaking this American Edition of Dr. Enfield’s Institutes oi 
-Natural Philosophy was to supply our Colleges with a book, which is held in so high estimation for 
the use of Students in recitations to their Instructers. In some of these seminaries of learning 
Ferguson’s Astronomy is recited, and in one or more of them after the exercises in Enfield’s Phi¬ 
losophy, exclusive of the astronomical part. It was the opinion of the Rev. Dr. Willard, President: 
of Harvard College, and Professor Webber, that it would be a valuable improvement in this part of 
Collegial Instruction to substitute Enfield’s Astronomy for Ferguson’s, provided some additions 
were made, which should be equally or more important than certain Articles in the latter, particu¬ 
larly those relating to the Calculation and Projection of Solar and Lunar Eclipses, including the 
necessary Tables. On account of some particular circumstances, measures were not taken to make 
the proposed additions till the printing was far advanced, and assurances given that it would be 
finished within a short period. Hence it was extremely difficult to execute the plan. Professor 
Webber, however, consented to attempt what then seemed to be practicable, and what, it was hoped, 
might in some good degree answer the purpose. And there appears to be a propriety in giving a 
particular account of the alterations that have been made. 

The errors, which occurred, are corrected. And it was thought expedient to retain the 
Introduction to the First Principles of Chemistry, but to annex it as an Appendix.* 

A change is made in the order of the branches by inserting Magnetism and Electricity between 
Pneumatics and Optics, instead of placing them after Astronomy; as it was thought a more natural 
and useful arrangement for a regular course of instruction, the propriety of closing with Astronomy 
being particularly obvious. 

In the astronomical part, the alterations are comprised in a few particulars, the substance oi 
which is almost entirely taken from Ferguson’s Astronomy, as well as the figures to which they 
refer.f 

An appendix of about SO pages is subjoined to the Astronomical part, and constitutes the most 
distinguishing and important peculiarity of this edition. It contains the most useful Solar and Lunar 


* See note p. ix. 

f The particular additions are the explanations of the circles of perpetual apparition and occultation under Prop. XX.— 
Scholium 2 to Prop. LX.—Scholium to Prop. LXXXIl.—Scholium to Prop. CXIV.—Explanation of the figures of orbits of 
Satellites under Scholium to Prop. CXVI. Scholium to Prop. CXL1II.—Cor. to Prop. CLXX11I.—Cor. to prop. CLXXV. 
A small alteration is made in the demonstration of Prop. CXIX. The Scholium to Prop. CXIX. and Scholium 3 to Prop. 
CXX. are omitted. The figures, together with four projections of eclipses, which are also added with the Appendix to the 
Astronomical part, fill two plates. 



ADVERTISEMENT. 


xii 

Tables, and a Table of Logistical Logarithms, together with their explanation, and twenty-two 
Problems, illustrating their use and application, and exemplifying the projection of Eclipses. These 
are selected from a work of Mr. Alexander Ewing, Teacher of Mathematics at Edinburgh, entitled 
“Practical Astronomy,” and published in 1797. With respect to the Projection of Eclipses, there 
is a small alteration in making two distinct Problems for the Lunar and Solar Projections, instead 
of placing them under the respective Problems for calculating those Eclipses. And two notes are 
added to the Problem for projecting a Solar Eclipse. The former contains the necessary directions 
for a different mode of projection ; and the latter the Rev. President Willard’s method of finding 
the point on the sun’s limb, where a Solar Eclipse begins; the knowledge of which point is of great 
importance to observers. 

In the whole execution of the work it has been the unfeigned endeavour of the editors to merit 
the approbation of the public. 

Boston, January , 1802. 


a 

ADVERTISEMENT 

TO THE THIRD AMERICAN EDITION. 

The following work has been, we believe, for some years, a classic in every College in New 
England, but considerable complaints having lately been made of its incorrectness and deficiency, 
it was expected that some other system would be brought forward to supply its place. None how¬ 
ever has appeared, and in the mean time the former editions have been taken up, and copies were 
continually called for, which could not be furnished, so that the public necessity seemed absolutely 
to require another edition to be immediately put to the press. Such corrections, however, have 
been made as the time permitted, and they will be found much more considerable, both in number 
and importance, than those in any former edition. The principal alterations occur in Book II. 
Props. 52, 57 and 80: B. III. P. 16 and 51; B. IV. P. 4; B. VI, P. 13,21,22, 23, 55, 56, 61, and 147; 
B. VII. P. 7, 36, 39, 45, 60, 70; Prob. at the end of Chap. IV. P. 80, 115, 136 ; Table at the end 
of part I. P. 167 and 170. In the Appendix, Explanation of Tables I. II. and VI, and the exam¬ 
ples under Problems 8 and 15. Less considerable alterations occur in every part of the work, too 
numerous to be particularized. These corrections have been made by the direction of James 
Dean, A. M. A. A. S. of Windsor, Vt. It is not pretended that the work even now is rendered 
perfect. The articles of Electricity and Magnetism, especially, are m tch behind the present state 
of science, and Physical Astronomy, also, is too concise and obscure; but* even as it has hitherto 
been, it seems to be the almost unanimous opinion of instructers, that it is better adapted to the 
state of science in this country than any other work extant; and the publishers can venture to 
assure the public that the present edition is very much improved. 

Boston, January , 1820. 




advertisement 


TO THE FOURTH AMERICAN EDITION. 

In correcting this fourth edition, the publishers have availed themselves of the minute and able Re¬ 
marks on the third, contained in the third volume of Silliman’s American Journal of Science and 
Arts, page 125. Of the corrections there suggested, those were made which would not cause 
a considerable change in the text; and the remainder are here subjoined. 

Book 1. Prop. JO. “ Some bodies appear to possess a power the reverse of the attraction of 
cohesion, called repulsion.” Of the five experiments adduced in support of this proposition, the 
first four,—namely, the depression of mercury around iron, and in capillary tubes, the suspension 
of a needle by water, and the depression of the surface around a floating piece of tin-foil,—are so 
far from furnishing any evidence of a repulsive power in bodies, that they are mere examples of 
cohesion, modified by circumstances. If we suppose the force of aggregation between the particles 
of mercury to be more than twice their force of cohesion to iron and to glass,* it appears from the 
investigations of Clairaut, that a depression ought to be the consequence.—The suspension of a 
small needle on water is owing to a certain degree of viscidity in this fluid,—in consequence of 
which the particles of the uppermost stratum present more resistance to separation than can be 
overcome by the downward pressure of the needle. Those who are acquainted with the extended 
researches of Count Rumford on this subject, will find no more reason for ascribing the support of 
a needle on water to a positive repellency, than the support of a cannon ball on ice. Both are 
alike owing to the cohesion of the upper surface. The only difference is, that as the cohesion 
is many times the least in the former case, the weight of the supported body must be proportionally 
less, when compared with the surface it exposes. 

Book II. Prop. A. This proposition, (besides that the demonstration is unsatisfactory,) is out 
of place ; as the chapter is confined by its title to the comparison o [uniform motions. It ought to 
have been deferred to ch. v. 

Prop. 22. Cor. 1. is evidently erroneous. A second “ Cor. 1.” is inserted under this proposition, 
which belongs to the preceding one; for it is true only of non-elastic bodies. 

Prop. 37. The demonstration is defective, from not being extended, as it easily might be, to the 
supposition that motion is lost in passing from one plane to another. The demonstration of the 
38th is also inconclusive, because it has not been previously shown that the total loss of motion in 
passing through a set of planes becomes evanescent, when the planes become indefinitely numerous, 
and their successive inclinations indefinitely small. 

Prop. 47. The demonstration shews that we may form a body by assembling particles round a 
given point , such that the body shall balance itself about this point; but it by no means shews that 
when the body is given, a point about which it will balance itself can be found ;—much less that 
this point, as the proposition implies, is the same for all positions of the same body. 

Prop. 49. The diagram employed in the proof of this proposition is drawn so inaccurately as to 
render it scarcely intelligible. There was the less reason for this inaccuracy, as in Rutherforth, 
from whom the diagram is copied, it is drawn correctly. 

Prop. 51. The demonstration of this important theorem is less general than the enunciation re¬ 
quires, by being confined to the case in which the bodies move in the same plane. The statement 
with which the first corollary begins is true only under such limitations as the student can scarcely 
be supposed able to apply. 

Prop. 56. In the great majority of instances in which the screw is employed, the resisting force 
is not moved up through an inclined plane, as the demonstration supposes. It would be far more 
simple and satisfactory to infer the law of equilibrium directly from the relative velocities of the 
points of application of the power and resistance. 

* Perhaps it ought rather to be said,—to “ the/dm of vioislure which is ordinarily attached to the surface of glass.” See 
Hatty— Traite de Physique, I. 225. Biot—I. 455. 


XIV 


ADVERTISEMENT. 


Prop. 57. Schol. 1. “ In all compound machines there will be an equilibrium, when the sum of 

the powers are to the weight, as the velocity of the weight is to the sum of the velocities of the 
powers.” No interpretation can be put upon this statement which will render it true. The error 
arose, we presume, in some such manner as the following. It was apparent that in compound ma¬ 
chines, (or rather in machines where several powers put several resistances in equilibrio,) the sum of 
the products of the powers each into its velocity, was equal to the sum of the products of the weights 
each into its velocity. This equation had the appearance of being capable of resolution into an 
analogy; and the resolution was accordingly made. But in doing it, two things were confounded 
which are widely different: the sum of the products, and the product of the sums. —It would have 
been better if the compiler had not attempted to deviate from Kutherforth. Supposing only one 
power, and only one resisting force which balance each other through the intervention of a series of 
mechanical powers,—the power will be to the weight simply as the velocity of the weight is to the 
velocity of the power.* 

Prop. 60. Schol. The method here given of finding the initial velocity of a projectile, gives only 
that part of it which is in a direction perpendicular to the horizon. To obtain the whole velocity, 
this result ought to have been increased in the ratio of radius to the cosecant of the angle of eleva¬ 
tion. But it would have been altogether preferable to omit noticing a method so entirely useless 
and even deceptive in practice, and to have substituted for it that by the ballistic pendulum. 

Prop. 68. In the last edition, several palpable errors which formerly perplexed the demonstra¬ 
tion have been corrected ; but the reasoning is still far from being demonstrative. The erroneous 
figure of former editions is also retained. The circle GNV, instead of GML, should have had T 
for its centre, and GML should have been an ellipse having T for its farther focus. 

Def. 16. Schol. It is improperly asserted in this scholium, that “ the projectile and the centri¬ 
fugal forces ditfer from each other as the whole from the part.” These forces are dissimilar in 
kind, and are incapable of comparison. They stand to one another in the same relation as pressive 
and percussive forces. If the tangent which measures the projectile, and the subtense which mea¬ 
sures the centrifugal force, be diminished indefinitely, as they must be before we can properly make 
the attempt to compare them, the latter becomes evanescent in respect to the former. The centri¬ 
fugal is rather a consequence of the projectile force, than a part of it. 

Prop. 70. “When bodies revolve in a circular orbit about a centre, the centripetal and centri¬ 
fugal forces are equal.” Thi3 proposition implies that in other orbits these forces are not equal. 
But the demonstration is such as would prove them no less equal, in all cases whatever.—We must 
confess ourselves at a loss to assign any consistent meaning to the term “ centrifugal force,” in re¬ 
lation to orbits not circular. Is this force measured by the distance by which a revolving body 
w’ould be more remote from the centre, in a given small time, if the centripetal force were suspend¬ 
ed, than it actually is while the centripetal force acts? If so, the centripetal and centrifugal forces 
are always equal, for the same point of an orbit, whatever be its figure. Or is it measured by the 
absolute increase of distance from the centre, which would take place in a given small time, if the 
body were abandoned to its projectile force? If so, in passing from the higher to the lower apsis 
of an eccentric orbit, the centrifugal force is a negative quantity. 

Lemma 4. The inference concerning the equality of the arc to the chord and tangent in their 
vanishing state, is inconclusive when the tangent is less than the arc, as it will be in certain positions 
of the subtense. The demonstration may be rendered complete in the following manner :— AB : 
AD i DB :: AC : AB-f-BC. But AB-fBC is ultimately equal to AC ; hence AD-fDB is ultimate¬ 
ly equal to AB, and much more is the arc AB equal to the chord AB. 

The first Part of Book III. which treats of Hydrostatics, presents us with several instances of 
explicit or implied error; particularly prop. 2, prop 13, Schol. and propositions 24, 26, 31, and 
36. But we have no room to dvveli upon them; and shall therefore pass directly to the second 
Part, which is devoted to Pneumatics. 

Prop. 51. The force with which wind strikes a sail of given dimensions, was stated in former 
editions as varying in the duplicate ratio of the cosine of the angle of incidence. In the last edition, 
the term sine is substituted for cosine . The phrase “ angle of incidence” was before used in the 
same sense as in optics : it is now employed in the ordinary sense of mechanics. But this correc- 

+ See Rutherforth’s System, Vol. I. Art. 72. 


ADVERTISEMENT. 


xv 


tion goes but little way towards freeing the proposition from exception. If the sail be supposed 
confined to move in the same direction with the wind, which the demonstration seems to imply, a 
resolution of the force on a third independent account was necessary ? which would have reduced 
that part of the force which is effective to the ratio of the cube of the sine, instead of the square.* * * § 
But as the variation of the force determined from experiment differs totally both from the square 
and the cube, it would have been belter to erase the proposition than to attempt to amend it. 

Prop. 55. Schol. The mode in which the constant velocity of sound is* attempted to be ex¬ 
plained, (which, like the rest of the scholium, is copied from Rovvning,) is wholly erroneous : nor 
do we think it easy to substitute an unexceptionable theory of the mechanism of aerial pulsations, 
without involving mathematical principles of a higher order than the student is supposed to be ac¬ 
quainted with. 

Prop. 56. Schol. 3. “ It is found by experiment, that air is necessary to the existence of sound, 

of animal life, of fire, and of explosion.” This, like several other statements scattered up and down 
the work, in which chemical principles are alluded to, needed correction to render it accordant with 
the present state of our knowledge on these subjects. The experiments of Biot and Chladni shew 
that sound is transmitted by solid bodies as well as gaseous ones, and that it may be conveyed to 
the organs of hearing without the intervention of air, by forming a communication between the 
sounding body and the head by means of a solid conductor.—That fire and explosion require air for 
their existence, is true only in the most loose and popular sense of the terms. In particular, that 
the explosion of gunpowder cannot be effected in a vacuum, as is implied in one of the annexed 
experiments, is an entire mistake.f 

The articles on the barometer, thermometer, hygrometer, and steam-engine, are extremely de¬ 
fective in point of valuable information, compared with what might have been said about them 
within the same limits, and in several respects are calculated to leave erroneous impressions: but 
we must content ourselves with giving this general caution against placing implicit confidence in 
them J 

Book IV. Prop. I. Scholium. The editor of the London edition of 1799, left the question un¬ 
decided, as Mr. Cavallo had done, whether any other than ferruginous bodies are capable of mag¬ 
netic properties. This scholium has been retained ; although the experiments of Tourte of Berlin,§ 
and Lampadius, place it beyond a doubt, that nickel and cobalt possess the same capacity, in this 
respect, with iron, except in an inferior degree. According to Lampadius, (see Thomson’s Annals, 
1815,) the “magnetic energies” of iron, nickel, and cobalt are as the numbers 55, 35, and 25 re¬ 
spectively. 

Prop. 2. Schol. If it were worth the while to retain the experiments of Musschenbroek on the 
variation of the magnetic force at all, in place of the far more important ones of Coulomb, the num¬ 
bers ought at least to have been corrected. Had tbe editor of the second edition taken these ex¬ 
periments from Musschenbroek himself, instead of taking them from Cavallo, he would have avoided 
the error of making the distances all ten times too large.|| The denomination employed in the 
original statement (see Philos. I. 206. Ed. 1744,) is tenths of inches, instead of inches. 

Prop. 10 To illustrate the mode of finding the declination of the needle by amplitudes of the 
heavenly bodies, the following example is stated : “ If the magnetic amplitude is 80° eastward of 
the north, and the true amplitude is 82° towards the same side, then the variation of the needle is 
2° west.” This statement cannot be reconciled to any definition of the term “amplitude;” and it 
cannot be reconciled with the usual definition and the one given in the Astronomy, except by 
making all the alterations which follow : “ If the magnetic amplitude be 10 J N. of E. and the true 
amplitude is S° towards the same side, then the variation of the needle is 2 J east.” 

*This theoretical determination may be seen, Gregory’s Mechanics, I 539, &c. 

f Robins, Hutton’s Math Dictionary, &c. 

j It maybe proper just to state, for the information of those who may have access to no other rule than that given page 
75, m making loose estimates of altitudes from the barometer, that the ascent corresponding to 1-10 in. fall of the mercury, 
instead of being one hundred and three feet, is at a mean, (that is, when the barometer itself is at 30 in. and the thermom¬ 
eter at 6o°.) only about ninety-three feet. 

§ Nicholson's Journal. Vo! 25 

|| If perfect exactness in such a case were of any importance, it would be necessary to recollect, that Musschenbroek’^ 
denominations were Rhinlaud measure ; which are greater than the English in the latio of 1,03 to 1. 


XVI 


ADVERTISEMENT. 


Prop. 10. Schol. 2. The late and accurate observations of Mr. Gilpin* and Col. Beaufoy on 
the diurnal variation of the needle in different months of the year, present wide deviations from the 
results here given from Mr. Canton. Both the observers just mentioned, make tbe extremes of the 
mean diurnal variation in different months, about 11' and 4'; and both place the time of the maxi¬ 
mum earlier in the year than was done by Mr. Canton. Col. Beaufoy (Thomson’s Annals, 1819,) 
places it in April. 

Prop. 11. Schol 1. The dip of the needle is here represented as probably “ unalterable at tbe 
same place.”—Whatever be the cause of the dip, this supposition is extremely improbable, while 
tbe declination is known to be variable, and to be, in common with the dip, the result of the ten¬ 
dency by which the needle places itself in the magnetical line. Nor do the observations made at 
London during the last century, warrant the inference made in this Scholium. As measured by 
Whiston in 1724, it was 75° 10 : and nearly accordant with this result is that of Graham, obtained 
in the following year. Cavendish, in 1775, found it to be 72° 30', and Gilpin, in 1805, 70 J 21'. 
These observations, after every allowance is made for the imperfection of the instruments employ¬ 
ed, leave no doubt that the inclination of the needle has undergone a gradual diminution in Lon¬ 
don, during the last century. According to M. Humboldt, (see Biot—Traite de Physique, HI. 136,) 
a similar diminution has taken place, during the same period, in France. 

Book VI. Prop. 13. Schol. 1. The statements concerning the ratio of the sine of the angle of 
incidence to that of deviation, in passing to and from water and glass, are true only under a limita¬ 
tion which is not distinctly pointed out,—namely, that the angle of incidence is indefinitely small. 

Schol. 2. The partial reflection of light by the second surface of transparent media, when the 
angle is within the limit for light to be refracted, is erroneously ascribed to “ inequalities” of the 
surface. If this were the true cause, no distinct image of an object could be seen by light thus 
reflected. 

Prop. 17. Exp. The effects of a single dense medium, bounded by a convex surface, on paral¬ 
lel, diverging, and converging rays, can never be illustrated by a convex lens, which produces two 
successive refractions,—one by a convex surface of the denser, and the other by a concave surface 
of tbe rarer medium. The lens presents the combined result of the former part of prop. 17, and 
the latter part of prop. IS. In particular, a convex lens can never render converging rays “less 
converging,” as is asserted in the fourth paragraph under the Exp. 

Precisely similar remarks might be repeated concerning the introduction of the concave lens to 
illustrate the several cases of prop. 18.—Both these experiments, if introduced at all, should have 
been placed after prop. 18; and the manner in which each illustrates both propositions should 
have been pointed out.f 

Prop. 22. Cor. 2. The corollary is right; but the investigation which is given of it, is incorri¬ 
gibly wrong. By comparison with the figure, it will be seen that it gives the position of the princi¬ 
pal focus of a glass sphere within the sphere ; and that of a sphere of water, coincident with the 
hinder surface. The proper mode of proceeding would be, first to determine the focus of parallel 
rays entering a denser medium by prop. 22 ; and then to find by prop. 23, the focus of rays con¬ 
verging (to the point just found,) when passing out of a denser medium into a rarer, through a con¬ 
cave surface of the rarer. 

Prop. 26. “ The image will not be distinct, unless the plane surface on which it is received be 

placed at the distance of the principal focus of the lens.” For “ principal,” read—“ corresponding 
to the distance of the object.” 

Prop. 35. “ Though the distance of the object from tbe lens be varied, the image may be pre¬ 

served distinct without varying the distance of the plane surface which receives it.” The distance 
of the plane surface from what ? Tne second mode of doing it, pointed out in the demonstration, 
is inconsistent with the supposition that the distance of the plane surface, either from the object, or 
tbe Jens, remains unaltered. Those who will consult Rutherforth’s Optics, Ch. VII. will see that 
this inconsistency has arisen from an attempt to blend into one, two propositions of which the con¬ 
ditions were different. We will add, although the remark has no relation to the last edition, that 
the mistake in the statement of the magnifying power of the double microscope (prop. 147.) arose 

* Philos. Trans. 1816. 

t Both these propositions have materially suffered in point of clearness, from employing as diagrams sections of lenses, 
instead of media indefinite in the direction towards which the rays proceed after refraction,—as well as from the inaccurate 
manner in which some of the lines are drawn. 


ADVERTISEMENT. xvi't 

from precisely the same source. Rutherforth investigated the two ratios on which the magnifying 
power depends in separate propositions,—first supposing the eye at the station of the object glass, 
and then at the limit of distinct vision. In uniting these two propositions into one, Enfield inad¬ 
vertently retained the condition of the former. 

Prop. 44. “ Reflection is caused by the powers of attraction and repulsion in the reflecting 

bodies.” This proposition is altered and abridged from the following in Rutherforth : “ Bodies 
refract and reflect light by one and the same power, differently exercised in different circum¬ 
stances.” The illustration of this proposition by the original author is an excellent one, consider¬ 
ing the state of optical knowledge at the time he wrote ; but in the hands of his abridger, although 
all the suppositions made by Rutherforth are retained, and we are required to admit that “ bodies 
attract those rays which are very near them, and repel those a little farther from them,” yet no use 
is made of the attracting surface, and the most interesting part of the proposition, the reflection 
produced by the second surface of the medium, (in regard to which so much pains had been taken 
in the previous scholium to exclude other hypotheses,) is entirely omitted. The student is left to 
wonder why “ attraction” is mentioned in the proposition as having any concern with reflection ; 
and the identity of action in the medium by which refraction and reflection are produced, is kept 
out of his sight. 

Prop. 46. Schol. Although perhaps nothing positively erroneous is advanced in this scholium 
concerning Sir Isaac Newton’s theory of fits of easy transmission and reflection, we cannot but 
object to a naked statement of a theory, stripped of all the facts which it was formed to explain, and 
made at the same time in so obscure a manner as must impair the respect of the student for its 
illustrious author. The hypothesis of fits, however it may seem fitted to excite ridicule as exhibited 
in this scholium, is now justly regarded as one of the most striking displays which Newton ever 
made of his transcendant genius. In the hands of Biot and his companions in the career of dis¬ 
covery, it has acquired an importance of which Newton himself could have had no adequate con¬ 
ception.—Whether the principles of this now highly interesting and important department of Optics 
can he reduced to the level of a system of elementary instruction, is deserving of serious inquiry. 
A digest of the phenomena and laws of polarization, involving no difficulties which would render it 
inaccessible, or deprive it of its interest with those who aim at nothing more than general views of 
science, appears at least to be as yet a desideratum. 

Prop. 58. “ In all mirrors, plane or spherical, &c.” This proposition, in regard to spherical 

mirrors, is true only of those pencils of reflected light which are indefinitely near the perpendicular. 

Prop. 69. In the demonstration it is stated, that “ by prop. 31, the diameter of the image, when 
the object is given, is inversely as the distance of the object.” This is not said, in prop. 31 ; nor 
is it true, except when the object is very remote. The image formed by a lens is not in circum¬ 
stances analogous to that produced on the retina of the eye; for the lens has no provision for pre¬ 
serving the image distinct, for different distances of the object, without varying the distance of the 
plane surface which receives it. 

Prop. 73. “ When equal objects in the same right line are seen obliquely, their apparent lengths 

are inversely as the squares of their distances from the eye.” The limitation “ in the same right 
line,” has been very properly inserted by the editor of the present edition ; but to render it correct, 
it wants another limitation which the proposition originally had as given by Rutherforth ; that is, 
“ When equal objects are seen very obliquely &ic.” When the object is of finite magnitude, the obli¬ 
quity must be very great, in order that the proposition may hold true,—unless indeed the object itself 
be very small; in which case it holds true for every degree of obliquity. But under this last modifi¬ 
cation, it requires a different demonstration ; and is more properly referred to the subject of appa¬ 
rent velocity than of apparent magnitude. As referred to the head of apparent velocity, the propo¬ 
sition might have been thrown into the following simple and not inelegant form : “ When a body 
moves uniformly in a right line, its apparent velocity will be inversely as the square of the distance 
from the eye.” 

In demonstrating the 83d and 85th propositions, it is stated as the reason why the image produced 
by a convex or concave lens is erect, that the axis of the pencils which proceed from the extremi¬ 
ties of the object “ only cross one another at the lens.” It should be, “ because they only cross 
one another at the eye.” The pencils which pass from the extreme points through a lens, do not, 
in fact, meet each other till they reach the eye. Figs. 8 and 9 convey no idea ol the manner in 
which the pencils come to the eye, except in the single case in which the eye is in contact with 

c 


XV111 


ADVERTISEMENT. 


the lens; nor is there any other diagram in the Optics which gives the student any information on 
this important point.—The remark scarcely need be added, that almost all the propositions in this 
chapter which state the effect of lenses on apparent magnitude, have unsatisfactory demonstrations. 
It is taken for granted that at whatever distance from the lens the eye is placed, the pencil which 
enters it from the same point of the object diverges as if from the same point in space. But the 
fact is, that as the eye recedes from the lens, the rays which enter the pupil frotn the same point of 
the object, gradually change : the axis of the pencil, instead of coinciding with* the centre of the 
lens, passes above or below it, according as the point of the object is above or below. Hence it is 
improper to assume that the pencil from A (figs. 8 and 9) diverges as if from the same point D for 
all distances of the eye from the lens. The assumption is erroneous, except when the object is 
extremely small, and it ought not to be made even in this case without proof.* 

Prop. 89. If this proposition were one of the least value, it would be desirable that it should 
have a more satisfactory demonstration than its present one, which on several accounts is wholly 
inconclusive. 

The statements concerning the brightness of the image, made in different propositions of this 
chapter, are not legitimately proved ; for the number of rays received by the pupil from any one 
point of the object may be increased, and the brightness nevertheless diminished,—on account of 
the increase of apparent magnitude. 

Props. J08, 110, and 111, assert unconditionally concerning the magnifying power of mirrors, what 
is true only in certain positions of the eye. If, for example, the object be nearer a concave mirror than 
its principal focus, and the eye be in the centre of concavity, the image, instead of “ appearing 
larger” than the object, as is asserted in prop. 108, will appear of the same magnitude ; and if the 
eye be brought still nearer the mirror, the image will appear the smallest. 

Prop. 144. Schol. 1. “ Of two refracting telescopes which magnify equally, the shorter will 

give a more imperfect image than the longer. For the image appeari: g equal in both, but being 
farther from the object-glass in the longer than the shorter, must be in reality larger or more mag¬ 
nified ; whence the defect arising from the different refrangibility of the rays, will be more visiule 
in the longer than in the shorter telescope.”—The statement with which this paragraph begins is 
correct. The reasoning subjoined is evidently erroneous, and leads to a conclusion the reverse of 
what was first asserted. If two telescopes were exactly similar in all their parts, differing only ia 
size, it is manifest that the imperfection of the image arising from unequal refrangibility, would be 
the same in both. But the smaller would have the disadvantage of rendering the object less bright, 
in the duplicate ratio of the linear dimensions. To render the brightness the same in both, the 
object glasses must be made equal; in which case the one of least focal distance, being a greater 
portion of a sphere, would produce the most imperfect image. 

Schol. 2. TIip account of achromatic lenses in this scholium omits the essential circumstance on 
which the whole explanation turns. We are told that a convex lens of crowm glass is to be united with 
a concave one of flint glass in such a manner that “ the excess of refraction in the crown glass may 
destroy the colour caused by the flint glass ” Here the student will naturally inquire, how the crown 
glass can possess an excess of refraction, without also possessing an excess of dispersive power? 
For the removal of this difficulty, no hint is given of the great fact which lies at the foundation of 
Dollond’s improvements, viz. that the dispersive power of different media is not proportioned to 
their mean refractive power. Unless he has the sagacity to conjecture that this may be the case 
from the obscure statement above quoted, he will remain in ignorance of what has been justly re¬ 
garded as the greatest discovery made in Optics since the period of Newton. 

Book VII. Prop. 13. To make the demonstration from fig. 10, consistent, EPLH ought to be 
regarded as a circle in the heavens ; it is therefore improper to place the spectator at P. The 
diagram should have been constructed like fig. 2, with a small concentric circle to denote the earth. 

Prop. 35. Cor. “ Hence it appears that the earth, at the winter solstice or Capricorn, is in its 
perihelion.” The student will be apt to infer, from this mode of expression, that the two points 
mentioned have some necessary connexion. But so far is this from being true, that the time when 
the earth is in its perihelion is about ten days later than that of the winter solstice. The angular 
motion of the earth in the interval (for 1820) is about 9° 50. 

♦The remarks made in this paragraph are equally applicable to the propositions in the succeeding chapter, which relafc 
to vision as affected by mirrors. 


ADVERTISEMENT. 


xix 


Prop. 35. Prol). 6. The method of finding the bearing of two places on the earth’s surface, here 
described, is manifestly erroneous, except when the places are very near each other. This part of 
the problem does not appear capable of a solution on the artificial globe. 

Chapter m. on Twilight, has undergone several material improvements in the last edition. The 
Cor. to prop. 37, is however out of place, and should have been expunged. The demonstration of 
prop. 39, is freed from several theoretical errors; although we think the attempt to distinguish be¬ 
tween the sun’s centre and upper limb, in an angle liable to so much uncertainty as the sun’s de¬ 
pression at the commencement of twilight, attended with no advantage sufficient to compensate for 
the additional complexness it gives the demonstration.—After all, we should have been much bet¬ 
ter pleased to see the proposition entirely omitted, than any attempt made to amend it. The hy¬ 
pothesis, that the rays which come to the eye at the end of twilight are brought by a single reflec¬ 
tion, is a very questionable one. The power of reflecting light possessed by the atmosphere, must 
depend on one or both of two causes : 1. It may reflect some of the rays which pass through it in 

consequence of a defect of transparency. 2. It may reflect in the same manner as light is ordina¬ 
rily bent back into a denser medium. This last mode of reflection, if it ever takes place without 
an abrupt change of density, is evidently more likely to take place, in proportion as the variation of 
density is more rapid. Now whichever of these causes operates to produce twilight, it must evi¬ 
dently exist in a far higher degree in the lower, than in the higher regions of the atmosphere. 
Hence instead of a single reflection at the height of forty-two miles, two or more successive reflec¬ 
tions may quite as probably transmit to the eye the light with which twilight closes.*—But even 
admitting the correctness of the assumption that twilight is produced by a single reflection, it is most 
obvious that no inference can be adduced concerning “the height of the atmosphere,” or even the 
height at which it ceases to reflect light. The only legitimate conclusion is, that forty-two miles is 
the limit beyond which light is not reflected in sufficient quantity to affect the organs of vision. If, 
instead of this vague proposition, the law of atmospheric density at different altitudes had been in¬ 
serted in its proper place in the Pneumatics, the subject would have been exhibited in a far more 
interesting and instructive form. 

The subject of the moon’s librations, in props. 78—82, is managed with singular infelicity. The 
introductory proposition should be, that “the time of the moon’s rotation on its axis is equal to the 
mean time of its revolution round the earth,”—instead of beginning with the fact that “the moon al¬ 
ways has nearly the same side towards the earth,” and drawing the strange inference that “if the moon 
revolves about its axis, its periodical time must be equal to that of its revolution round the earth.” 
The librations should be assigned each to its proper cause ; that in latitude to the obliquity of the 
axis to the plane of the orbit, and that in longitude to the eccentric form of the orbit,—instead of 
blending the explanations of both under the loose proposition, “ the librations of the moon may be 
exolained on the supposition that the moon has a revolution on its axis.” In prop. 81, the equality 
of the times of rotation and revolution is inferred from the librations ; while it is in fact a matter of 
direct observation, and must be presupposed in explaining the librations themselves. In prop. 82, 
the elliptical form of the moon’s orbit is inferred from the libration in longitude. We very much 
doubt whether the species of oval to which the moon’s orbit most nearly approaches, could have 
been determined from direct observations on so trifling a change of phase. The proper mode of 
presenting this part of the subject would be, to infer the elliptical form of the orbit from the observed 
relation between the anomaly and the apparent diameter; and then to employ this conclusion for 
the explanation of the libration in longitude. 

Pr<fp. 83. Schol. In stating the results of Dr. Herschel’s observations on the altitude of the 
lunar mountains, it is mentioned, that “ one was found to be about a mile in height; but none of 
the others which he measured proved to be more than half that altitude.” By consulting the origi¬ 
nal memoir in the Philos. Trans, and another which he has published since, it will be seen that 
Dr. Herschel’s results differ much less from the estimates of the older Astronomers, and from the 
recent and accurate measurements of Schrbter, than is here represented. Dr. H. makes several 
over a mile, and one, nearly two miles in height. 

Prop. 10G. “If the moon, when new, is in one of its nodes, the eclipse of the sun will be cen¬ 
tral.” it should be,—“ to the inhabitants of some part of the earth it wili be centrally eclipsed in 
the zenith.” To those parts of the earth at which the moon is never vertical, a central eclipse can 
happen only when the moon is not in its node, at the time of conjunction. 

* See Vince’s Ast. 1. Art. 206. 


XX 


ADVERTISEMENT. 


Prop. 113, which affirms a motion “ in antecedentia,” of the satellites of the superior planets 
while passing from one elongation to the next through their inferior conjunction, is no less errone¬ 
ous than the propositions of former editions concerning the retrogradation of the primary planets ; 
and should, like them, have been rectified or struck out. All that can be said with truth is, that 
during the specified interval the motion of the secondary is retrograde relatively to its primary ; and 
even this statement can scarcely be extended to the satellites of Herschel.* 

Prop. 114. “ The greatest elongations of a satellite on each side are equal ” This proposition 

has several exceptions. The orbits of the third and fourth satellites of Jupiter have a very sensible 
eccentricity; and the same is true of the fourth (now more generally numbered as the sixth) satel¬ 
lite of Saturn. See Laplace : Syst. du Monde. The latter, according to Delambre, (Ast. III. 510,) 
has an ellipticity nearly equal to that of our moon. 

Prop 123. Several of the particulars inserted in the annexed scholium from Sir I. Newton, 
have now become obsolete. In particular the quantities of matter in Jupiter and Saturn, instead of 
being to that in the sun in the ratios of 1 to 1100 and 2360, are now known to be in the ratios of 
1 to 1067 and 3534.f 

Prop. 135 is founded on the erroneous theory of retrogradation previously laid down; and there¬ 
fore should have been corrected. 

Prop. 155. The demonstration of this proposition is in part fallacious. It is said to be contrary 
to prop. 51. cor of Book II, that the centre of gravity of two gravitating bodies should move ; and 
is inferred that if one of the bodies is projected in any direction, the other must acquire (by what 
means we are not told) an equal motion in the opposite direction. Now this is so far from following 
as a necessary consequence, that the other body will not in fact acquire any such motion ; and if a 
projectile movement be given to one ol them alone, the common centre of gravity of the two will 
not continue at rest. Nor does this contradict the proposition referred to in the Mechanics ; for 
the common centre will move uniformly in a right line. The proposition should have stood thus : 
“ The sun and any planet revolve round a common centre of gravity, which remains at rest, or has 
a uniform rectilineal motion.” 

Prop. 162. This theorem, as it stood in Rowning, was preceded by an investigation of the 
motion of the apses produced by a force varying in a greater or less than the inverse duplicate ra¬ 
tio of the distance. As nothing analogous to this investigation has been retained by Enfield, the 
assertion that when the force varies faster than in the inverse duplicate ratio of the distance the line 
of the apses will move fonvard, and vice versa, made in the course of the demonstration, is wholly 
gratuitous. 

Prop. 163. The demonstration is not only irrelevant to the proposition, but from an inadvertent 
change in the conditions as laid dowm by Rowning, a blunder is carried through it and the annexed 
corollary. The demonstration affirms, that if the moon is passing from the higher to the lower 
apsis and its gravity increases too fast, “ it will approach nearer to the earth” than it would other¬ 
wise do, “ and describe a portion of an orbit less eccentric, or nearer a circle.” The former state¬ 
ment is correct ; but it contradicts the latter. So in the corollary we are told that “ when the 
gravity of the moon towards the earth decreases too fast, the eccentricity of the orbit will increase ; 
and when her gravity towards the earth increases too fast, the eccentricity will decrease.” The fact 
is, that in both cases alike the eccentricity will increase. It is when the gravitv increases or di¬ 
minishes too slow, that the eccentricity will decrease Those w 7 ho will give themselves the trouble 
of consulting the prop, as it stands in Rowning, wfill find no difficulty in perceiving how a hasty 
abridger might shift the conditions of the demonstration. % 

Props. 164 and 166. Why two propositions so nearly identical should find a place in this 
chapter we can give no account,—unless that the compiler had forgotten that he had given a theo¬ 
rem on the motion of the nodes from Rowning, and therefore looked for one in some other author. 

* We have not attempted to rectify the periodical times of the satellites of Herschel; for with the exception of the 
second and fourth, their distances from their primary are wholly conjectural; nor is even their number regarded by Dr. 
Herschel as yet fully ascertained His last determination of the synodical revolutions of the second and fourth, given in 
the Philos. Trans, for 1815, is as follows: 

II. 8d. 16h 56'5". 

IV. 13d 1 lh. 8' 59". 

The inclination of their orbits to the ecliptic lie finds to be 78° 58',—much farther from perpendicularity than has been 
heretofore supposed. 

f Mec. Celeste. Part II. Ch. 9. 


ADVERTISEMENT. xxi 

So much at least is certain,—that prop. 166, and this only, among those which compose the chap¬ 
ter, is borrowed from Rutherforth. 

Prop. 168. Schol. This method of finding the direction of gravity includes only the effect of 
the centrifugal force. Including the joint effect of rotation and of figure, the direction is manifestly 
that of a perpendicular to the tangent plane of the earth’s surface, or of a normal to the elliptical 
curve of the meridian passing through the given place. 

Prop. 173. In the concluding paragraph of the demonstration, the relative forces of the sun 
and moon to raise tides are erroneously stated. The real forces are directly as the masses, and 
inversely as the cubes of the distances. 

The concluding scholium of the Astronomy consists of extracts from a paper of Dr. Herschel’s in 
the Philos Trans, for 1795. These extracts are so unskilfully made, and are presented in so dis¬ 
jointed a form, as to afford scarcely any idea of the train of argument pursued in the original. But 
in the original itself, high as is the estimation in which the author is justly held as an observer, we 
must be permitted to think that there are several statements which cannot be defended. With the 
view of multiplying the points of analogy between the sun and the planets, and thus increasing the 
presumption that the former is inhabited, he endeavours to shew that both primaries and secondaries 
shine in some measure by their own light. The partial illumination of the moon, for example, 
during a total eclipse, cannot be entirely ascribed to the-light which may reach it from the earth’s 
atmosphere ;—“ because, in some cases, the focus of the sun’s rays refracted by the earth’s atmos¬ 
phere must be many thousand miles beyond the moon.” Dr. Herschel assumes as the basis of this 
calculation, that the rays of the sun are bent by the atmosphere at only an angle of 31'. He seems 
to have inadvertently neglected the circumstance that the rays undergo a second equal refraction 
in passing out of the atmosphere. In consequence of this, the real inflection is 62' ^or rather 66', 
taking 33' as the mean horizontal refraction,) so that the focus of the sun’s rays as refracted by the 
earth’s atmosphere can never in fact be so distant as the moon. An observer stationed at the 
moon, even during a central eclipse of the sun, would see a luminous ring encircling the earth. 
The light thus thrown upon the moon’s disc is amply sufficient to explain its partial illumination 
during a central eclipse. Were Dr. H.’s assumption concerning the amount of atmospheric refrac¬ 
tion correct, his conclusion would not follow; for the same agency of the atmosphere which pro¬ 
duces twilight to an observer stationed on the earth’s surface, will produce the same effect to a 
second spectator, stationed any where behind the first, and in the same tangent plane of the earth.* 
Another obvious proof that Dr. H. was misled by his zeal to find points of analogy between the 
sun and the other bodies of the system, at least so far as the phosphorescent quality of the moon is 
concerned, is, that light is not given off in any sensible degree from the crescent which is unen¬ 
lightened bv the sun, just before and after opposition. 

The attempt to remove an objection to the sun’s being inhabited by supposing that “ heat is 
produced by the sun’s rays only when they act on a calorific medium,” and that they are the 
cause of heat only “ by uniting with the matter of fire which is contained in the substances that 
are heated,” together with the arguments advanced in support of these strange positions, certainly 
ought, for the credit of one who has deserved so highly of astronomical science, to have been sup¬ 
pressed. They are too far behind the present state of Chemistry, and too little essential to the object 
which their author had in view, to deserve transcribing into the pages of an elementary work, 
which is intended to be employed in instruction. 

In passing to the Appendix ,—our limits will not allow us to notice a variety of errors which occur 
in the progress of the examples ; nor a number of small inaccuracies unnecessarily introduced into 
the mode of projecting solar eclipses. The tables of epochs (which terminate with the present 
year) should have been extended ; and might also have been advantageously corrected from those 
of Delambre and Burckhardt.—But the most important positive error, perhaps, which occurs in the 
Appendix, relates to the method of finding the arguments of the moon’s latitude. In Ewing’s As¬ 
tronomy and all the editions of Enfield except the last, we have given, over the I lid table, “Arg I— 
1>’s mean anomaly and over the Vth, “ Arg. IV—q's mean anomaly.” Both these arguments 
are wrong. Those who may have the curiosity to look into Mason’s edition of Mayer’s Lunar Ta- 

* Hence the 93d Proposition, which ascribes the light transmitted to the moon’s disc during a total eclipse <o “ the 
reflection of rays of light falling upon the earth’s atmosphere,” is doubtless in part correct 


xxii ADVERTISEMENT. 

hies, from which Ewing’s were abridged, will see at a glance how these erroneous captions originat¬ 
ed. They are in fact the 3d and 5th arguments of Mayer’s tables; but Mayer’s 3d table is omitted, - 
and his 10th is made Ewing’s 5th. The captions were inadvertently copied, although they belong¬ 
ed, in consequence of these omissions, to the wrong tables. In the last edition, the caption of the 
third table is altered to make it agree with the general directions for finding the arguments of Lati¬ 
tude given in Prob. 8th ; but that of the 5th still remains erroneous, as well as the general rule under 
Prob. 8th. It should be, “ subtract the moon’s mean anomaly from the second argument,” &c.; 
and the caption of table 5th should be, “ Arg. 11 — ’s mean anomaly.” 

The principal part of the corrections and alterations made by the editor of the last edition have 
our entire approbation. Particularly in regard to two highly important propositions, the one relating 
to the law of refraction, in the Optics, and that on the sun’s parallax, in the Astronomy, he has pro¬ 
bably done the best that the elementary character of the work admitted. There are a few instances, 
however, of alterations, the propriety of which appears very questionable, and which justice to the 
labours of former editors requires us briefly to notice. 

Thus in the first proposition, “ Matter may be, and mere extension is infinitely divisible,” the 
clause in italics is peculiar to the last edition. We recollect having seen in Hutton’s Dictionary an 
attempt to establish a distinction between ‘‘actual” and “potential divisibility but we could not 
understand it; nor are we any more fortunate in regard to the language just quoted. If the term 
“ divisibility” itself means nothing more than the possibility of being divided, to say that matter 
may be infinitely divisible is a solecism. The distinction between the divisibility of matter and 
that of extension seems to depend on the definition of the term. If by “ divisibility” be meant 
merely the possibility of being ideally divided by mathematical planes without any separation of 
parts, then the property is one which belongs to matter and to extension in precisely the same 
sense. But if in the phrase “ divisibility of matter” be included the additional idea of discerptibili- 
ty, or the possibility of being separated into parts not in contact, then the property is one which be¬ 
longs in no degree to pure extension. In neither case does the distinction made in the proposition 
as quoted above appear to have any foundation. That matter is infinitely divisible in the first sense, 
is almost self-evident: whether it is so in the last, (admitting the exercise of any supposable power 
which does not change the nature of matter,) is a question which lies beyond the reach of the hu¬ 
man faculties. 

We notice, in the second place, that three experiments, on the approach of light bodies floating 
on water to each other, or to the side of the vessel, have been transferred from prop. 5, where they 
were originally placed to illustrate the cohesion between solids and fluids, to prop. 4, where, if they 
illustrate any thing, it must be the cohesive attraction between two solid bodies. It is true that these 
phenomena are only indirect consequences of the attraction between solids and fluids ; and a scho¬ 
lium was very properly added by the author, (which has been omitted in all the subsequent edi¬ 
tions,) to aid the student in tracing their connexion with the proposition. But it is most certain 
that they have nothing to do with the cohesive force of two solid bodies. When there is an eleva¬ 
tion or depression of the fluid around both of two floating bodies, they will approach : when there 
is an elevation around one and a depression around the other, they will recede. These are mere 
results of capillary action ; and as such, admit of an easy explanation from the general theory of 
Laplace.*—A popular idea of the mechanism of these phenomena will be gained from the following 
experiment, by which we have been much amused, and which we do not recollect to have seen 
noticed. Two small globules of mercury, carefully laid upon water, will swim. Let these globules 
be brought within one or two inches, and it is surprising to observe the rapidity with which they 
dart together. If one of the globules is forced to the edge of the water, (the vessel being of such 
materials as to be capable of being moistened,) it will recede with an activity which might seem the 
effect of animation. But on holding the vessel in the light, the secret of these motions will be ap¬ 
parent. Each globule will be seen to have a depression around it, which perceptibly extends to the 
distance of more than half an inch'. The globules will be seen to rush together, not from any mu¬ 
tual attraction, but because, in doing it, each descends down an inclined plane. Two needles, laid 
on water and kept parallel to each other, will exhibit similar appearances. 

In the Optics, under Prop. 13. Exps. 22, 23, 55, 5G, &c. the term focus, as used to denote the 

* See Mec. Celeste. Sup. au dix. Livre : Biot—Traite de Physique, I. 462.: Hatty I. 237. 





ADVERTISEMENT. 


xxm 


point as iff rom which diverging rays proceed after refraction or reflection, is changed into imagina¬ 
ry radiant. The latter term is doubtless tlie most descriptive of the actual condition of the rays, 
and by some writers is uniformly employed instead of virtual or negative focus. But to intr oduce 
this distinction increases the complexness of enunciation of several important theorems which are 
already too complex.* It were to be wished, for the sake of these theorems, that we had some 
term which should merely express the point where the lines of direction of a pencil of rays meet, 
before refraction or reflection,—without including the idea of divergency or convergency; and 
another to denote the same thing after refraction or reflection. As long as this is not the case, we 
are not confident that any advantage is gained by changing the denomination of focus, when virtual, 
to imaginary radiant . But if the change is made at all, it ought at least to be carried through. 
This has not been done by the Editor: and the consequence is, that several propositions contain 
an implied error. He has inserted “ imaginary radiant” after “focus” in prop. 55; but in props. 
22, 23, and 56, which equally required a siindar addition, and in prop. 54, which required a sub¬ 
stitution, neither has been made. Such an addition would, it is true, have rendered the enunciation 
of some of these propositions exceedingly perplexed ; but consistency demanded that it should be 
done, or that the language of former editions should be left unaltered.f 


The publishers have also to acknowledge their obligations to the learned gentleman by whom 
the third edition was so much improved, for suggesting some additional corrections. These partly 
relate to points noticed in the preceding extract from Silliman’s Journal. Such of them as were 
received in season, were adopted in the text. [See Cor. 4. Prop. xxvi. B. II. (transferred from 
Prop xii. B. vi.)—Problem, page 182—Problems viii. and xv. in the Appendix.] The rest of 
them follow here. 

Add to the Scholium, Prop. 36, B. II, the following sentence. “ When the body descends in a 
curve line, the deflection being less than any assignable angle, its cosine may be reckoned equal to 
radius, and the retardation as nothing.” 

Let Prop. 45. B. II. be omitted, and the 46th, with the two paragraphs which contain its demon¬ 
stration, take its place ; and let the one given below, with its demonstration and corollary, be in¬ 
serted as Prop. 46. The schol. under Prop. 45, may follow the corollary given below as Schol. 
1, while the two scholia under Prop. 46, may be numbered 2 and 3 respectively. 

“ Prop. 46. If a pendulum be made to vibrate in a cycloid, the time of a vibration, whether the 
arc be long or short, will be to the time in which a body will fall freely through half the length of 
the pendulum, as the circumference of a circle to its diameter. 

PI. 2. Let a body descend in a cycloid from B to X, and it will acquire the same velocity as a 
F ' s 4 ' body falling freely from D to X (by Prop. 36), that is, a velocity which, if uniformly continued 
during the time of the fall from D to X, would carry it through twice that distance (Prop. 27.) or 
BX. (Lem. 5. cor ) But a body moving along BX tends towards X with a force every where pro¬ 
portional to its distance therefrom (Lem. 6.); and the time of describing a line with a motion accele¬ 
rated by such a force, from whatever point of the line it commence its motion, or half the time of a 

*Such in Enfield’s Optics, are props. 21, 23, 54, 56. 

f It must be admitted that the language of former editions, in this respect, was not entirely consistent with itself. Defs. 
8 and 18, and the Schol. to def. 13, needed modification. 


XXIV 


ADVERTISEMENT. 


vibration (by Prop. 38.) : the time of describing the same line with the velocity acquired at X 
continued uniformly through the line, or the lime of falling through DX, half the length of the 
pendulum :: half the circumference of a circle : its diameter. And, doubling the antecedents, the 
whole time of a vibration : the time of falling through half the length of the pendulum :: the 
whole circumference of a circle : its diameter. 

Cor. From this analogy the lengths of pendulums and the spaces described by falling bodies 
may be compared, and either of them computed from the other. For since the spaces described 
by falling bodies are as the squares of the times, reckoning both from the beginning of the descent 
(Prop. 26.), then the square of the time of one vibration of a pendulum : the square of the lime in 
which a body would fall through half its length :: the square of the circumference of a circle : the 
square of its diameter : : 9.8690 : 1 : : the space described by a falling body in any given time 
from the beginning of its fall : half the length of a pendulum which performs a vibration in the 
same time. Or inversely (as the length of a pendulum and time of its vibration are more easily 
and more accurately ascertained by experiment than the velocity of falling bodies,) 1 : 9.8696 : : 
half the length of a pendulum : the space through which a body will fall while it performs one 
vibration.” 

On page 234, omit lines 26-30, and let the next paragraph begin with “ The” instead of 
“ Another.” 


ERRATA. 




V fK ( r? 

I 


Page 

83 

Line 

15 

from the top, for 

to the magnet , read of the magnet. 

ioo 

12 

— bottom, — 

zinc , — 

the zinc. 

10S 

2 

— top, — 

it , — 

if- 

125 

15 

— bottom, — 

distance, — 

distances. 

166 

19 

— top, — 

latitude , — 

altitude. 

167 

16 

— — — 

15" one minute, — 

1 5' one minute. 

187 

7 

— bottom, — 

is will , — 

it will. 

193 

3 

— — — 

1 58, — 

0 58. 

■ 

v 0 

ti <4i 

M7 J ' v - 

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v t i tt : \ i 

et f t •:* 


CONTENTS 


BOOK I. 

Page. 

OF MATTER. 1 

BOOK II. 

OF MECHANICS, OR THE DOCTRINE OF MOTION. 

Chap. I. Of the General Laws of Motion, 7 

Chap. II. Of the Comparison of Uniform Motions, - - - 8 

Chap. III. Of the Composition and Resolution of Forces, - - 10 

Chap. IV. Of Motion as communicated by Percussion in Non-Elastic and 

Elastic Bodies, - - - - - 12 

Chap. V. Of Motion as produced by the Attraction of Gravitation, - 15 

§. 1. Of the Laws of Gravitation in Bodies falling without Obstruction , Ibid. 

2. Of the Laws of Gravitation in Bodies falling down Inclined Planes, 18 

§. 3. Of the Pendulum and Cycloid, ----- 21 

§. 4. Of the Centre of Gravity, - - - - - 27 

Chap. VI. Of Motion as directed by Certain Instruments called Mechanic¬ 
al Powers, - - - - - 29 

Chap. VII. Of Motion as produced by the United Forces of Projection and 

Gravitation, - - - - - 35 

§. 1. Of Projectiles, - - - - - Ibid. 

2. Of Central Forces, - - - - - - -38 


BOOK III. 

OF HYDROSTATICS AND PNEUMATICS. 

PART I. 

OF HYDROSTATICS. 

Chap. I. Of the Weight and Pressure of Fluids, 48 

Chap. II. Of the Motion of Fluids, - - - - 54 

§. 1. Of Fluids passing through the Bottom or Side of a Vessel, - - Ibid. 

2. Of Rivers, ------- 56 

Chap. III. Of the Resistance of Fluids, ----- 58 

Chap. IV. Of the Specific Gravities of Bodies, - - - - 60 

Table of Specific Gr avities, - - * - - - 64 

d 




XXVI 


CONTENTS. 


PART II. 


OF PNEUMATICS. 

Chap. I. Of the Weight and Pressure of the Air, 65 

Chap. II. Of the Elasticity of the Air, - - - - 67 

Of the Syphon, and Syringe, - 72 

Of the Common Pump, the Forcing Pump, the Condenser, and 

the Air-Pump, - - - - 73 

Of the Barometer, ------ 74 

Of the Thermometer, - - - - - 75 

Table of Heat, ------ 76 

Of the Hygrometer, - - - - - Ibid. 

Of the Steam Engine, - - - - - 77 

Of the Hydrometer, - - - - - - 78 


BOOK IV. 

OF MAGNETISM. - - 79 


BOOK V. 

OF ELECTRICITY. - - 88 


BOOK VI. 

OF OPTICS, OR THE LAWS OF LIGHT AND VISION. 
Chap. I. Of Light, ------- 

Chap. II. Of Refraction, - - - - - 

1. Of the Laws of Refraction, - 

2. Of Images produced by Refraction, - 

Chap. III. Of Reflection, ------ 

1. (f the Laws of Reflection, - 

2. Of Images produced by Reflection, - 

Chap. IV. Of Vision, ------ 

($. 1. Of the Laivs of Vision, ------ 

2. Of Vision as affected by Refraction, - 

3. Of Vision as affected by Reflection, - 

Chap. V. Of Colours, - 

1. Of the different Refrangibility of Light, - 

2. Of the Rainbow, ------ 

Chap. VI. Of Optical Instruments, - 

1. Of Telescopes, - 

^.2. O/' Microscopes, - 

3. D/* the Magic Lantern, - 

4. Of the Camera Obscura, - 


102 

104 

Ibid. 

115 

119 

Ibid. 

125 

126 
Ibid. 

131 

135 

140 

Ibid. 

144 

149 

Ibid. 

153 

154 

155 





CONTENTS. xxvii 

BOOK VII. 

OF ASTRONOMY. 

PART r. 

OF THE MOTIONS OF THE HEAVENLY BODIES. 

Chap. I. Of the Solar System in General, - - . 156 

Chap. II. Of the Earth, ------- 157 

1. Of the Globular Form of the Earth, and its Diurnal Motion about its 

Axis, and of the Appearances which arise from these , - - Ibid. 

2. Of the Annual Motion of the Earth round the Sun, - - 160 

3. Of Twilight, - - - - - - -172 

4. Of the Equation of Time, - - - - - 174 

Chap. Ill. Of the inferior Planets, Mercury and Venus, - - - 1/8 

Chap. IV. Of the Superior Planets, Mars, Jupiter, Saturn, and the Herschel, 181 
Chap. V. Of the Moon, - - - - - - -183 

1 . Of the Variations in the Appearance of the Moon, - - Ibid. 

2. Of Eclipses, - - - - - - -187 

Chap. VI. Of the Satellites of Jupiter, Saturn, and the Herschel, - 194 

Chap. VII. Of Comets, - - - - - 197 

Chap VIII. Of the Sun, ------- 199 

Chap. IX. Of the Parallaxes, Distances, and Magnitudes of the Heavenly 

Bodies, ------- 200 

PART IT. 

OF THE CAUSES OF THE CELESTIAL MOTIONS AND OF OTHER PHENOMENA. 

Chap. I. Of the Cause of the Revolutions of the Heavenly Bodies in their Orbits, 208 
Chap. II. Of the Lunar Irregularities, - - - - - 211 

Chap. III. Of the Spheroidical Form of the Earth, - - - 216 

Chap. IV. Of the Precession of the Equinoxes, - - - 2l7 

Chap. V. Of the Tides, ------- 219 

PART III. 

OF THE FIXED STARS. - - . 221 

APPENDIX TO THE ASTRONOMY, 

Containing Solar and Lunar Tables, with their explanation and use, and the Projection of Eclipses, 
selected from “ Ewing's Practical Astronomy.” 

Explanation of the Tables, ------ 233 

Problems, showing the use and application of the Tables, and Projection of Eclipses, 239 
Solar and Lunar Tables, ------ 263 


V 










* 































¥ 











P 
















BOOK 1 


f 


OF MATTER. 

Definition I. Matter is an extended, solid, inactive, and moveable substance. 

Scholium. Extension and solidity are discovered to be properties of matter by the senses. Both by 
the sight and touch we perceive material substances to have length, breadth, and thickness, that is, to be 
extended: and from the resistance which they make to (he touch, we acquire the idea, and infer the 
property of solidity. It is unnecessary here to inquire, whether solidity necessarily supposes impene¬ 
trability. Natural Philosophy, being employed in investigating the laws of nature by experiment and 
observation, and in explaining the phenomena of nature by these laws, has no concern with metaphys¬ 
ical speculations, which are generally little more than unsuccessful efforts to extend the boundaries of 
human knowledge beyond the reach of the human faculties. 

Def. II. A body is any portion of matter. 

Corollary. All bodies have some figure ; for, being portions of matter, they are finite, and there¬ 
fore bounded by lines either straight or curved. 

PROPOSITION I. 

Matter may be, and mere extension is, infinitely divisible, or capable of being di¬ 
vided beyond any supposed division. 

1. Any particle of matter, placed upon a plane surface, has an upper and a lower side, or a part which 
touches, and another which does not touch the plane, and is therefore divisible. 

2. Let CO, MD, be two parallel right lines, to which let AB be drawn perpendicular. In the line Plate 1. 
MD, on one side of the perpendicular AB, take, at equal distances, the points E, F, G, H. On the other Fig. 1. 
side of AB, in the line CO, take any pc,' it C, and join CE, CF, &,c. Each of the lines CE, CF, &c. will 

cut off a portion from AB : But whatever number of lines be drawn in the same manner from C to MD 
produced, there will still remain a portion of AB not cut off; because no line can be drawn from the 
point C to the line MD, which shall coincide with CO : The line AB is therefore infinitely divisible. 

3. Let the right lines AC and GH be drawn perpendicular to the right line BF. In AC, produced Plate 1. 
at pleasure, take any points C, C, &c. from which, as centres, with the distances CA, CA, &c. describe Fig. 2. 
arcs of circles KAL, NAO, &c. touching BF in the point A, and cutting HG. The farther the central 

point is taken from A, the greater will be the circles, and the nearer will the arcs approach to the line 
BF; but (El. III. Pr. 16. Cor.) the arcs, touching BF in A, cannot touch it in any other point. The line 
HG is therefore infinitely divisible. 

From this proposition the following theorems are derived by Dr. Keill, in his fifth 
lecture. 

Theorem i. Any quantity of matter, however small, and any finite space, however large, being 
given, (as for example, a cube circumscribed about the orb of Saturn,) it is possible for the small quan¬ 
tity of matter to be diffused throughout the whole space, and to fill it so that there shall be no pore or 
interstice in it, whose diameter shall exceed a given finite line. , 

Cor. Hence there may be given a body, whose matter, being reduced into a space absolutely full, 
that space may be any given part of its former magnitude. 



2 


OF MATTER. 


Book I. 


Theorem ii. There may be two bodies of equal bulk, whose quantities of matter, being unequal, in 
any proportion; yet the sum of their pores, or quantity of void space in each of the two bodies shall be 
to each other nearly in a ratio of equality.—Example. Suppose 1000 cubic inches of gold when reduced 
into a space absolutely full, to be equal to one cubic inch: then 1000 cubic inches of water, which is 19 
times lighter than gold, will, when reduced, contain -J^th part of an inch of matter. Consequently the 
void spaces in the gold, will be to those in water as 999 to 999if, or nearly in the ratio of equality. 

In the present state of knowledge, it is impossible to determine how far the division of matter can 
be actually carried, or whether there be any indivisible atoms by the arrangement and combination of 
which all sensible bodies are formed. We are, however, furnished both by art and nature with many 
astonishing instances of minute division.—If a pound of silver and a grain of gold be melted together, 
the gold will be equally diffused through the whole silver; and if a grain of the mass, containing only 
the 5761th part of a grain of gold, be dissolved in aqua fortis, the gold will subside. 

A grain of gold may be spread by the gold-beater into a leaf containing 50 square inches, and this 
leaf may be divided into 500,000 parts : and by a microscope, magnifying the diameter of an object 10 
times, and its area 100 times, the 100th part of each of these, that is, the 50 millionth part of a grain of 
gold, will be visible. 

The natural divisions of matter are still more wonderful. In odoriferous bodies a surprising subtilty 
of parts is perceived : several bodies scarcely lose any sensible part of their weight in a great length of 
time, and yet continually till a very large space with odoriferous particles. Dr. Keill has computed the 

magnitude of a particle of assaftetida to be only the , ,000,000,00^000.000,000 th part ° f a CuMc inch ' 

Again, Mr. Leewenhoeck informs us that there are more animals in the milt of a cod-fish than there are 
men on the whole earth, and a single grain of sand is larger than four millions of these animals. More¬ 
over, a particle of the blood of one of these animacula has been found, by calculation, to be as much 
smaller than a globe of the -i-th of an inch in diameter, as that globe is smaller than the whole earth. 
Nevertheless, if these particles be compared with the particles of light, they will be found to exceed 
them in bulk as much as mountains do single grains of sand. 

These instances may serve to show the amazing fineness of the parts of bodies, which are never¬ 
theless compounded. Gold, when reduced to the finest leaf, still retains those properties which arise 
from the modifications of its parts. Microscopic animalcula are, without doubt, organized bodies, and 
the globules of their blood are possessed of specific qualities. Even the rays of light are compounded 
of an indefinite variety of particles, which, when separated, have the power of exciting the ideas of 
colours. 

Def. III. That force by which the parts of the same body, or of different bodies, 
on their contact, or near approach, are united to or tend towards each other, is called 
the attraction of cohesion. 

PROP. II. The attraction of cohesion appears in solid bodies. 

Exr. 1. Observe the different degrees of cohesion in different kinds of wood, suspending weights, 
from pieces of equal diameter, placed vertically or horizontally till the}' break. 

2. Measure the different degrees of cohesion in silk, thread, horse-hair, kc. by weights suspended 
by cords of each, placed vertically or horizontally. 

The result of sundry experiments, made by professor Musschenbroek, to show the cohesive power of 
different solids, may be seen in the following table. In estimating the absolute cohesion of solid bodies, 
he applied weights to separate them according to their length: the pieces of wood which he used were 
parallelopipedons, each side of which was -^-ths of an inch, and the metal wires made use of were T Lth 
of a Rhinland inch in diameter, and they were drawm asunder by the following weights : 



lb. 


lb. 

Fir .... 

600 

Copper - 

2991 

Elm .... 

950 

Brass .... 

360 

Alder - 

1000 

Gold .... 

500 

Oak - - - 

1150 

Iron .... 

450 

Beech .... 

1250 

Silver .... 

370 

Ash .... 

1250 

Tin .... 

Lead - 

491 

291 


From the experiments of Guyton Morveau, the following are the utmost weights, which wires of 
0.787 of an English line in diameter can support without breaking: 



Book I. 


OF MATTER. 


3 


lb. Av. 

A wire of Iron supports * 549.25 

Copper - •. 302.278 

Platinum. 274.32 

Silver.187.137 

Gold. 150.753 

Zinc .109.54 

Tin.34.63 

Lead . 27.621 

PROP. III. The attraction of cohesion takes place between particles of the same 
fluid. 

Exp. 1. A drop of water, at the end of a small cylinder of wood, will hang in a spherical form 
The drop is spherical, because each particle exerts an equal power in every direction, drawing other 
particles towards it on every side as far as its power extends. 

2. Two globules of mercury, on meeting, unite, which union can arise only from their strong attrac¬ 
tion. Drops of water will do the same. 

PROP. IV. The attraction of cohesion takes place between two solid bodies of the 
same kind ; and the more perfect the contact, the greater is the attracting force. 

Exp. 1. Two glass bubbles, floating near each other on water, rush together. 

2. A glass bubble floating on water in a glass vessel moves towards the side of the vessel. 

3. Two circular pieces of cork, placed upon water, and brought near each other, will be attracted. 

4. Two plates of glass laid together, though perfectly dry, will cohere. 

5. Two leaden balls, having each a flat surface of a quarter of an inch in diameter, scraped smooth, 
on being forcibly put together, will cohere so strongly as sometimes to require a weight of nearly 1001b. 
to separate them. 

6. Two polished plates of brass, smeared with oil, will cohere strongly. 

M. Musschenbroek found that the adhesion of polished planes, about two inches in diameter, heated 
in boiling water, and smeared with grease, required the following weights to separate them: 

Cold Grease. Hot Grease. 


Planes of Glass 

1301b. 

3001b. 

Brass 

- 150 

800 

Copper - 

200 

850 

Marble 

- 225 

600 

Silver 

150 

250 

Iron - 

- 300 

950 


PROP. V. The attraction of cohesion takes place between solids and fluids. 

Exp. 1 . A plate of glass, or metal, will retain drops of water, or mercury, when inverted. 

2. If a plate of glass be in part immersed in a vessel of water, the water which lies contiguous to 
the glass will rise above the level. 

3. Water rises above its level between two parallel plates of glass at a small distance from each 
other, and in a glass tube having a fine bore, called a capillary tube. 

4. The fluid will rise between parallel plates, and in capillary tubes, in vacuo. Hence it appears, 
that the ascent of fluids in capillary tubes is not owing to the pressure of the air. 

5. Human blood will rise to a great height in a tube having an exceedingly fine bore. 

6. Water will ascend in the cavities of sponge, sugar, and other porous bodies. 

7. If a drop of oil be poured upon a plate of glass laid horizontally, and another plate of glass be so 
placed as to meet the first plate at one edge, and be at such a distance from it at the other, as just to 
touch the drop of oil; this drop, because its touching surface is continually enlarging, will move, with 
increasing velocity, towards that edge. If the planes be lifted up on the side where they meet, the 
motion will be retarded, stopped, or reversed, according to the degree of elevation. 

8. The same phenomenon takes place in a tube of unequal bore. 

9. A circular piece of ice, two inches and a half in diameter, exactly balanced and brought to touch 
the surface of some mercury, will be so strongly attracted by the mercury, as to require more than nine 
pennyweights, in the opposite scale, to restore the equilibrium. 

10., A piece of wood having a smooth and plane surface, suspended from a beam and balanced, on 
touching a surface of water, will be attracted; and it will require an additional weight in the opposite 
scale to separate them. 


4 


OF MATTER. 


Book I. 


Scholium 1. As it is by the attraction of cohesion that the parts of a body are kept together; so 
when a body is broken, this attraction is only overcome. Hence the reason of the soldering ot metals, 
gluing of wood, &c. Hence also may be explained why some bodies are hard , others so/’t, and others 
fluid , which properties may result from the different figures of the particles, and the greater or less 
degree of attraction consequent thereupon. Elasticity may arise from the particles of a body, when 
disturbed, not being drawn out of each other’s attraction; as soon therefore, as the force upon it ceases 
to act, they restore themselves to their former position. 

Scholium 2. Solids are dissolved in menstruums from the particles of the solid being more attracted 
to the fluid than to themselves. Precipitation arises from a like cause ; for if to the solution of any 
solid in a fluid, some other solid or fluid be added, the particles of which are attracted by the fluid with 
a greater force than those of the solid which was dissolved, the solid falls to the bottom in a fine pow¬ 
der. Thus silver dissolved in aqua fortis is precipitated by copper. 

PROP. VI. The heights to which a fluid arises between parallel plates of glass 
are inversely as the distances of the plates. 

The absolute attractive force of the plates will always remain the same, whatever be the distance 
of the plates. The same weight of fluid must, therefore, at different distances of the plates, be*sup¬ 
ported. But the quantity of fluid supported can only continue the same, when the height of the column 
supported is reciprocally as its base; that is, when as much as the height is increased the base is dimin¬ 
ished, and the reverse. Now, the length of the base remaining unvaried, the base can only be made 
greater or less, by increasing or diminishing the distance between the plates. Therefore, the force, 
and the quantity of fluid supported, remaining the same, the height will be greater as the distance ot 
the plates is less, and the reverse. 

Let H, B, D, express the height, base, and distance, when the plates are at any given distance, and 
A, 6, rf, express the same when they are brought nearer : from what has been shown, H : h :: b : B ; but 
b : B : : d : D ; therefore H : h : : d : D. 

Exp. Let two parallel plates of glass be immersed, at different distances from each other, in a vessel 
of coloured water. 

PROP. VII. The suspension of the fluid, in capillary tubes, is owing to the at¬ 
traction of the ring of glass contiguous to the upper surface of the fluid.* 

Every ring of glass below the surface attracts the water above it as much downwards, as it attracts 
the water below it upwards, and consequently can contribute nothing towards the support of the column: 
and the action of the lowest ring upon all the fluid in the tube, within its surface of attraction, must 
either concur with the force of gravity to bring the fluid downwards, or, acting upon it at right angles, 
can have no effect in suspending it within the tube. The fluid therefore can only be supported by the 
ring of glass contiguous to its upper surface, which, attracting upwards, opposes the action of gravitation 
by which the fluid endeavours to descend. This reasoning may be applied to the fluid raised between 
parallel plates of glass. 

Exp. Let a capillary tube be composed of two parts, the bore of one of which is wider than that 
of the other: immerse its wider orifice in water, till it is filled to any heigfit less than the length of the 
wider part; the fluid will only rise to the height to which it would rise if the tube were throughout of 
the same bore with the wdder part: but immerse the tube till the fluid enters the smaller part, and the 
whole column will be suspended, provided its length do not exceed that of the column which a tube of 
the smaller bore is capable of supporting. 

Hence it is manifest, that the water is sustained by the attraction of the narrower part of the tube, 
for the wider part could not sustain so long a column: it is also manifest, that it is sustained by the ring 
contiguous to the upper surface ; for if it were sustained by the ring at the lower surface, no reason 
could be assigned why this should now support the greater column in both parts of the tube, when it 
was before only able to sustain a column which filled a part of the wider tube. 

Next, let the tube be inverted, and the water be raised into the lower extremity of the wider part; 
when the suspended column is of greater length than that which a tube of the same bore with the wider 
part is capable of sustaining, it will immediately sink: whence it is manifest, that the suspension of the 
column in this case depends upon the attraction of the wider part of the tube ; for the narrower part 
could sustain a larger column: and also, that it is sustained by the ring contiguous to the upper surface ; 

* This proposition has been disputed. Dr. Hamilton, in his second lecture, supposes that the suspension arises from the 
attraction of the annulus lying just within the lower orifice of the tube. But Mr. Parkinson rejects both suppositions, and 
concludes, that the fluid is sustained by the immediate attraction of the glass. See Parkinson’s Hydrostatics, p. 39 


5 


Book I. OF MATTER. 

for if it were sustained by the ring at the lower surface, it has been seen that this ring could support a 
much longer column. 

Schol. The reason why the narrower or wider ring sustains a column of the same length in the 
unequal tube above described, as in a tube throughout of the same diameter as the upper ring, is that 
the moving forces of the columns are in both cases the same; as will be more fully shown hereafter. 

Book III. Pr. iv. Schol. 

PROP. VIII. In capillary tubes, the heights to which the fluid rises are inversely 
as the diameter of the bores. 

The fluid being suspended (Prop. VII.) by the ring of glass contiguous to the upper surface, and the 
distance to which the attracting force of glass reaches being unvaried; the attracting force which sus¬ 
tains the fluid will be as the number of attracting particles, that is, as the circumference, or diameter of 
the ring, or of the tube. Let Q,, 7 , then, represent the quantities of fluid to be raised in two tubes of 
different bores; D, rf, the diameters of their bores; and H, h, the heights to which fluids rise in the 
tubes ; because Q, 7 , represent two cylinders of the fluid, from the properties of the circle and cylinder 
(El. XII. 2. 11 , and 14) Q, : 7 :: DDH : ddh ; and from the nature of this attraction, which is as the diame¬ 
ters of the tubes, Q, : 7 : : D : d ; therefore DDH :: d dh : : D : d; and consequently D : d : : h : H. 

Cor. From this proposition it appears, that in any glass capillary tube, the height to which it will 
elevate water, and keep it sustained, multiplied into the diameter of the tube, is a given quantity; this 
is found by experiment to be .053 part of an inch ; by means of this value the diameter of a capillary 
tube being given, the height to which it will elevate water will be known, for it will be equal to .053, 
divided by the diameter; thus suppose the diameter is ^ of an inch, the height to which the water will be 
elevated = .053 x 20 = 1.06. 

Exp. Let two tubes of different bores be immersed in a vessel of coloured water; it will be found, 
that the water will rise as much higher in the smaller tube, as the diameter of its bore is less than that 
of the larger tube. 

PROP. IX. Between two glass plates, meeting on one side, and kept open at a 
small distance on the other, water will rise unequally; and its upper surface will form 
a curve, in which the heights of the several points above the surface of the fluid will 
be to one another reciprocally, as their perpendicular distances from the line in which 
the plates meet. 

Let AE be the surface of the fluid ; AF the line in which the plates meet; HL the curve formed Plate 1 . 
by the surface of the raised fluid; GB, IC, KD, LE, perpendicular to AE, expressing the heights of Fig. 3. 
the respective points G, I, K, L, in the curve, above the surface of the fluid, and AB, AC, AD, AE, per¬ 
pendiculars to AF, expressing the distances of the same points from the line in which the plates meet: 
these heights and distances are reciprocally proportional. For let the lines GB, IC, KD, LE, repre¬ 
sent pillars of fluid of an equal but exceedingly small breadth ; those portions of the glass plates, which by 
their attraction, support these pillars being equal, will sustain equal quantities of fluid; that is, the pil¬ 
lars will be equal. But the pillars may be considered as parallelopipeds, which (El. XI. 34) are equal 
when their bases and altitudes are reciprocally proportional. And the bases, being equal in breadth, 
are as their lengths, that is, as the intervals between the plates : and since' the intervals continually in¬ 
crease as the distance from the line in which the plates meet increases, these intervals, at the points 
B, C, D, E, are as their distances AB, AC, AD, AE, from the line AF. Since, then, the heights of the pil¬ 
lars are reciprocally as the intervals, the heights GB, IC, &c. are reciprocally as the distances AB, AC, 

&c. This is the property of an hyperbola, whose asymptotes are AE and AF. 

Exp. Let coloured water rise between two glass plates (their inner surfaces being first moistened) 
meeting on one side according to the proposition. 

PROP. X. Some bodies appear to possess a power the reverse of the attraction 
of cohesion, called repulsion. 

Exp. 1 . If a piece of iron be laid upon mercury, the surface of the mercury near the iron will be 
depressed. 

2. A fine needle laid upon water will sw im. 

3. Two circular plates of tinfoil being placed upon water, and pressed down by a smajl additional 
weight upon their surface, repelling the water, will have a cavity round them : but when they are 
brought near each other, they will rush together; the reaction of the water on the outer side of the 
plates being greater than the reaction on the inner side, where the two cavities produced by repulsion 
are united. 


6 


OF MATTER. 


Book I. 


4. Mercury, poured into a recurved glass tube, having the bore on one side exceedingly fine, and on 
the other large, will not rise so high in the narrow, as in the wide bore : water will rise higher.* 

5. Melted glass dropped into water, forms globules with a stem, (called Prince Rupert’s drops) 
which on breaking the stem will burst with great violence, and fall into powder. 

PROP. XI. All bodies on or near the surface of the earth tend towards its centre, 
by the attraction of gravitation. 

A stone or other heavy body, let fall, will move towards the earth till it meet with some other body 
to obstruct its course. And bodies move in lines perpendicular to the surface, because the point to which 
they ultimately tend is the centre of the earth, and the line of direction produced coincides with the 
radius, and is at right angles with the surface, which is nearly spherical. Some bodies ascend, because 
they are acted upon by a force greater than the attraction of gravitation, and in a contrary direction. 
Vapours, smoke, &c. do not descend, because they are lighter than the air, and supported by it. 

Exp. 1 . Smoke or steam will descend in an exhausted receiver. 

2. Any boiling fluid being placed in a scale and balanced, the balance will be destroyed by evapo¬ 
ration. 

Schoi.. 1 . When we speak of attracting powers, we do not attempt to explain their nature, or as¬ 
sign their causes. Having derived general principles, or laws of nature, from phenomena, we only give 
a name to these principles, in order to explain other appearances by them. 

Schol. 2. The tendency of all bodies towards the earth really results from their tendency towards 
the several parts of the earth. For by an experiment made by Dr. Maskelyne upon the side of the 
mountain Schehallien, he found the attraction of that mountain sufficient to draw the plumb-line sensibly 
from the perpendicular. See Phil. Trans. Vol. LXV. or Sir J. Pringle’s Discourses. 

♦The phenomena exhibited by the four preceding experiments are rather to be considered as examples of cohesion, modi¬ 
fied by circumstances. See American Journal of Science, &c. Vol. III. p. 133. 


/ 


BOOK II. 


OF MECHANICS, OR THE DOCTRINE OF MOTION. 

CHAPTER I. 

> 

Of the General Laws of Motion. 

PROPOSITION I. 

E VERY body will continue in its state of rest, or of uniform motion in a right line, 
until it is compelled, by some force, to change its state. 

Any tody at rest on the surface of the earth will always continue so, if no external force be im¬ 
pressed upon it to give it motion, and if the obstacle which hinders the attraction of gravitation from car¬ 
rying it towards the centre be not removed. A body being put into motion by some external impulse, if 
all external obstructions were removed, and the attraction of gravitation suspended, would move on 
forever in a right line ; for there would be no cause to diminish the motion, or to alter its direction. This 
cannot be fully established by experiment, because it is impossible entirely to remove all obstructions; 
but, since the less obstruction remains the longer motion continues, it may be reasonably inferred, that 
if all obstacles could be removed, motion once communicated to any body would never cease. 

Exp. 1. A body at rest requires some degree of force to put it in motion : and when in motion, it 
will continue to move longer on a smooth surface than on a rough one; instances of which occur in the 
use cf friction rollers; in the exercise of skating, &.c. 

2. If a stone be whirled round in a string, on being set at liberty it will continue to move with the 
force which it has acquired. 

3 . If a vessel containing a quantity of water be moved along upon a horizontal plane, the water, 
resisting the motion of the vessel, will at first rise up in the direction contrary to that in which the 
moving force acts : when the motion of the vessel is communicated to the water, it will persevere in 
this state ; and if the vessel be suddenly stopped, resisting the change from motion to rest, it will rise 
up on the opposite side. In like manner, if a horse which was standing still, suddenly start forwards, 
the rider will be in danger of being thrown backwards; if the horse stop suddenly, the rider will be 
thrown forwards. 

Sciiol. This proposition suggests two methods of distinguishing between absolute and apparent mo¬ 
tions. (1.)Absolute motion, or change of absolute motion, may sometimes be distinguished from appar¬ 
ent, by considering the causes which produce them. When tw o bodies are absolutely at rest, they are 
relatively so; and the appearance is the same when they are moving at the same rate, and in the same 
direction : a relative motion, therefore, can only arise from an absolute motion in one or both of the 
bodies, which (by the Prop.) cannot be produced but by force impressed. Hence, then, if we know that 
such a cause exists, and acts upon one of the bodies, and not upon the other, we may conclude that the 
relative motion arises from a change in the state of rest, or absolute motion of the former; and that 
with respect to the latter, the effect is merely apparent. Thus when a person on board a ship observes 
the coast receding from him, he know'S the appearance arises from the motion of the ship upon which 
the wind or tide is acting. 

(2.) Absolute motion may sometimes be distinguished from apparent by the effects produced. A bodk¬ 
in absolute motion endeavours to proceed in the line of its direction: if the motion be only apparent, 
there is no such tendency. It is in consequence of the tendency to persevere in a rectilinear motion, 
that a body, revolving in a circle, constantly endeavours to recede from the centre. This effort is cal¬ 
led a centrifugal force; and as it rises from absolute motion only, whenever it is observed, are con¬ 
vinced that the motion is real. 

Exp. Let a bucket, partly filled with water, be suspended by a string, and turned round till the 
string is considerably twisted ; then let the string untwist itself. At first the water remains at rest, but 





OF MECHANICS. 


Book II. 


as il acquires the motion of the bucket, the surface grows concave to the centre, and the water ascends 
up the sides, thus endeavouring to recede from the axis of motion ; and this effect increases till the wa¬ 
ter and bucket are relatively at rest. When this is the case, let the bucket be suddenly stopped and the 
absolute motion of the water will be gradually diminished by the friction of the vessel; and at length, 
when it is again at rest, the surface becomes plane. Thus the centrifugal force does not depend upon 
the relative, but upon the absolute motion, with which it begins, increases, decreases, and disappears. 

PROP. II. The change of motion produced in any body is proportional to the force 
impressed? and in the direction of that force. 

Effects are proportional to their adequate causes. If, therefore, a given force will produce a given 
motion, a double force will produce the double of that motion. If a new force be impressed upon a body 
in motion, in the direction in which it moves, its motion will be increased proportionally to the new 
force impressed: If this force act in a direction contrary to that in which the body moves, it will lose 
a proportional part of its motion : If the direction of this force be oblique to the direction of the moving 
body, it will give it a new direction. 

Exr. Let one clay ball, suspended by a string, strike another clay ball suspended in the same man¬ 
ner at rest or in motion, it will communicate a degree of motion greater or less in proportion to the 
force of the striking body : In the opposite direction, motion will be destroyed in the same proportion. 

Cor. Since the effect produced by two bodies upon each other, depends upon their relative velocity, 
it will always be the same whilst this remains unaltered, whatever be their absolute motions. 

PROP. III. To every action of one body upon another, there is an equal and con¬ 
trary re-action : Or, the mutual actions of bodies on each other are equal and in con¬ 
trary directions, and are always to be estimated in the same right line. 

Whatever quantity of motion any body communicates to another, or whatever degree of resistance it 
takes away from it, the acting body receives the same quantity of motion, or loses the same degree of 
resistance in the contrary direction: the resistance of the body acted upon producing the same effect 
upon the acting body, as would have been produced by an active force equal to, and in the direction of, 
that resistance. 

Cor. 1 . Hence it appears, that one body acting upon another, loses as much motion as it communi¬ 
cates ; and that the sum of the motions of any two bodies in the same line of direction cannot be changed 
by their mutual action. 

Cor. 2. This proposition will explain the manner in which a bird, by the stroke of its wings, is able 
to support the weight of its body. For if the force with which it strikes the air below it, is equal to 
this weight, then the re-action of the air upwards is likewise equal to it; and the bird being acted upon 
by two equal forces, in contrary directions, will rest between them. If the force of the stroke be greater 
than its weight, the bird will rise with the difference of these two forces: And if the stroke be less than 
its weight, then it will sink with the difference. 

Exp. Let a clay ball in motion strike another equal to it at rest: The striking body will lose half its 
quantity of motion, which will be communicated to the other body. 

Schol. These three laws of motion may be illustrated by experiments, but their best confirmation 
arises from hence, that all the particular conclusions drawn from them agree with universal experience. 
They were assumed b)' Sir Isaac Newton as the fundamental principles of mechanics ; and the theory of 
all motions deduced from them, as principles, being found to agree, in all cases, with experiments and 
observations, the laws themselves are considered as mathematically true. 


CHAPTER II. 

Of the Comparison of uniform Motions. 

PROP. IV. The quantities of matter in bodies are in the compound ratio of their 
magnitudes and densities. 

If the magnitudes of two bodies be given, the quantities of matter will be as the densities: If their 
densities be given, the matter will be as the magnitudes : therefore the matter is universally in the 
compound ratio of the magnitudes and densities. For example; If A and B be two balls equal in mag- 



Chap. II. 


COMPARISON OF MOTIONS. 


nitude, the quantity of matter in A will be to that in B, as the density of A is to that of B : if both be of 
the same density, their quantities of matter will he as their magnitudes. 

PROP. V. The velocities, with which bodies move, are directly as the spaces they 
describe, and inversely as the times in which they describe these spaces. 

It is manifest, that the degree of velocity increases as the space a body passes over in a given time 
increases, and as the time in which it passes over a given space decreases; and the reverse. For 
example ; If one ball A move over six feet, and another ball B over three feet in the.same time, A ha- 
double the velocity of B ; but if the ball A passes over six feet in two seconds of time, and the ball B 
passes over six feet in one second, the velocity of B is double of that of A. 

PROP. VI. The spaces which bodies describe are in the compound ratio of their 
times and velocities. 

It is evident, that the longer time any body continues to move, and the greater velocity it moves 
with, the greater space it will pass through ; and the reverse. If, for example, the body A move for 
one second, and the body B move for two seconds, both moving with the same velocity ; A will move 
through half as much space as B: If A move with two degrees of velocity, and B with one degree of 
velocity; A will, in the same time, pass over twice as much space as B. 

PROP. VII. The times in which bodies move are directly as the spaces, and 
inversely as the velocities. 

The greater space any body passes through, and the less degree of velocity it moves with, the 
greater must be the portion of time taken up in the motion ; and the reverse. For example ; If the 
ball A move with the same velocity with the ball B, but pass over double the space, A will move 
twice as long as B; If A move over the same space with B, and with half the velocity, it must, in this 
case also, move twice as long as B. 

PROP. A. If bodies be acted upon by different constant forces, the velocities com¬ 
municated will vary in a ratio compounded of the forces and times. 

Let F, V, T, represent force, velocity, and time, and be supposed variable ; it is evident that the 
velocity will be increased and diminished in the same ratio with both force and time, and these being 
independent of each other, V will be as F x T. 

Cor. If, therefore, F be compared with any other known force/ capable of generating a velocity 
equal to v in the time <, then V:v::FxT:/x(. 

PROP. VIII. The power required to move a body at rest is as the quantity of 
matter to be moved. 

Each particle of matter in any body resisting motion, a force must be exerted upon each particle to 
overcome this resistance ; if, therefore, two bodies containing different quantities of matter are to be 
moved, the greater body will require the greater force. 

Def. I. The momentum of any body is its quantity of motion. 

PROP. IX. In moving bodies, if the quantities of matter be equal, the momenta 
will be as the velocities. 

It is manifest, that if the body A be equal to the body B, but A have twice the velocity of B, A has 
twice as much motion as B. 

PROP. X. The velocities of two bodies being equal, their momenta will be as their 
quantities of matter. 

The bodies A and B moving with equal velocities, since every portion of matter in A has as much 
motion as an equal portion of B, it is evident, that if A have twice the quantity of matter in B, it must 
have twice as much motion. 

PROP. XI. The momenta of moving bodies are in the compound ratio of their quan¬ 
tities of matter and velocities. 

The greater quantity of matter there is in any body, and the greater velocity it moves with, the 


10 


OF MECHANICS. 


Book II. 


Plate 1. 
Fig. 4. 


greater will evidently be its quantity of motion; and the reverse. If, for example, the body A he 
double of the body B, and move with twice its velocity, the momentum of A w ill he quadruple of that 
ofB : For it will have twice the momentum of B from its double velocity, and also twice the momentum 
of B from its double quantity of matter. 

Cor. Hence, if in two bodies the product of the quantities of matter and velocities are equal, their 
momenta are equal. 

PROP. XII. The velocities of moving bodies are as their momenta directly, and 
their quantities of matter inversely. 

The greater momentum any body has, and the less quantity of matter it contains, the greater must 
he its velocity. For example ; If the body A be half of B, and their momenta be equal, A w ill move 
w ith twice the velocity of B ; and if A and B are equal, and the momentum of A is double of that of B, 
its velocity will also be double. 

PROP. XIII. The force, or power of overcoming resistance, in any moving body, 
is as its momentum. 

Since a body having a certain degree of motion is able to overcome a certain degree of resistance, 
it is manifest, that with an increased momentum, it will be able to overcome a greater resistance. 

Cor. Hence the momentum of any body is measured by its power of overcoming resistance. 

Schol. Let Q, q, denote the quantities of matter in any two bodies, D, (/, their densities, B, 6, 
their bulk or magnitude, V, d, their velocities, T, <, the times of their motion, S, s, the spaces over 
which they pass, P, p , the moving powers, M, m, their momenta , and Ftheir force. The preceding- 
propositions may be thus expressed : 


Prop. IV. 

Q, : 

<1 

: BD 

: b d 




S 

s 

V. 

V : 

V 






T 

7 

VI. 

S : 

s 

: TV 

t V 




S 

s 

VII. 

T : 

t 






V 

V 

A. 

V : 

V 

: F X T 

fxt 

VIII. 

P : 

p 

: Q : 

q 

IX. 

M : 

m 

: V : 

v if 

X. 

M : 

m 

: Q : 

q if 

XI. 

M : 

m 

: QV : 

q v 

XII. 

V : 

V 

M 

m 




: Q : 

- q 

XIII. 

F : 

f 

: M : 

m 


CHAPTER III. 

Of the Composition and Resolution of Forces. 

Def. A. Equable motion is either simple or compound. Simple motion is that 
which is produced by the action, or impressed force, of one cause. Compound motion is 
that which is produced by two or more conspiring powers, i. c. by powers whose di¬ 
rections are neither opposite nor coincident, 

PROP. XIV. A body acted upon by two forces united, will describe the diagonal 
of a parallelogram, in the same time in which it would have described its sides by the 
separate action of these forces. 

If in a given time, a body, by the single force M impressed upon it at a point A, would be 
carried from A to B; and by another single force N impressed upon it at the same point, would 
be carried from A to C; complete the parallelogram ABDC ; and with both forces united, the 







Chap. 111. 


COMPOSITION OF FORCES. 


11 


body will be carried in the same time through the diagonal of the parallelogram from A to I). For 
since the force N acts in the direction of the right line AC parallel to BD, this force (by Prop. II.) has 
no effect upon the velocity with which the body approaches towards the line BD- by the action of the 
force M. The body will therefore arrive at the line BD in the same time, whether the force N is im¬ 
pressed upon it or not; and at the end of that time will be found somewhere in the line BD. For the 
same reason at the end of the same time it will be found somewhere in the line CD ; therefore it must 
be found at the point D, the intersection of these two lines. And (by Prop. I.) it will move in a right 
line from A to D. 

Exp. Two equal leaden weights, suspended at the end of a triangular frame of wood to give them a 
steady motion, and let fall at the same instant from equal heights, striking a ball suspended by a cord at 
the point in which their lines of direction meet, will carry it forwards in the diagonal of the parallelo¬ 
gram of those lines produced. 

Cor. 1 . Hence, the velocity produced by the joint action of two forces is to that with which the 
body moves by the action of each force singly, as the diagonal of the parallelogram to either side; for 
the diagonal is described in the same time with either side. 

Cor. 2 . If two sides of a triangle represent the directions and quantities of two forces, the third side 
will represent the direction and quantity of a force equivalent to both acting jointly : For the third side 
may be considered as the diagonal of a parallelogram. 

Cor. 3. A body may be moved through the same line by different pairs of forces. In plate 1. tig. 4. 

AD is the diagonal both to the parallelogram A^DC, and to the parallelogram AEDF; and consequently 
expresses a force equal to AB, AC, and to AF, AE. 

Cor. 4. Hence we learn why any heavy body let fall perpendicularly from the top of a mast, when a 
ship is under full sail, will fall to the bottom, in the same manner as if it had been at rest. 

Scholium. This proposition may be farther illustrated. If two men sit upon the opposite sides of a 
■boat under full sail, and toss a balldo^each other, they will catch the ball in their turn, just as they would 
have done if the boat had been at rest. The ball is here acted upon by two forces : (1.) it partakes ot 
the motion of the boat, which is common to the ball, the boat, and the men : (2.) the other force is that 
with which the man throws it across the boat. By these two forces together, the ball will describe the 
diagonal of a parallelogram, one of whose sides is the line that the boat has described whilst the ball was 
flying across ; and the other side is a line drawn from one man to the other. 

PROP. XV. The velocity produced by two joint forces, when they act in the same 
direction, will be as the sum of the forces, and when they act in opposite directions, will 
he as their difference ; and the velocity will be the greater, the nearer they approach to 
the same direction, and the reverse. 

In the parallelograms ABCD, in which AB, AC, express the direction and quantity of two joint 
forces, the side AB, being placed at different angles with AC, it is manifest, that as AB approaches towards 
AC, the diagonal increases, till at length it becomes equal to AC+CD, that is, to AC-f-AB, and the veloc¬ 
ity is as the sum of the forces, since they act in the same direction. 

In the parallelograms ABDC, as AB recedes from CD, the diagonal increases, till at length it van- Plate 1. 
ishes with the angle, and the two sides AB, AC, constitute one right line, the parts of which, AB, AC, Fl g- 6 - 
representing forces acting in opposite directions, if the forces be equal, they will destroy each other; 
if unequal, the velocity will be as their difference. 

PROP. XVI. Any single force or motion may be resolved into two forces or mo¬ 
tions ; and the directions of these may be infinitely varied : also any two forces may be 
compounded into single forces. 

A body moving in the line AD, may be considered as receiving its direction and velocity from two Plate 1. 
forces acting jointly in the directions AB, AC, or from two other forces expressed by AF, AE . For F, g- 4. 
(Prop. XIV 7 . Cor. 3) each pair would produce the same effect. In like manner the direction and quan¬ 
tities of the forces will be diversified with every change of the sides of the parallelogram, the diagonal 
remaining the same. 

It is also manifest, that any two joint forces may be compounded into one, being expressed by the 
sides of a parallelogram, or its diagonal. 

PROP. XVII. If a body is acted upon by three forces, which are proportional to, 
and in the direction of, the three sides of a triangle, the body will be kept at rest. 

Let a body placed at D be acted upon by three forces AD. CD, FD, proportional to, and in Plate 1. 


12 


OF MECHANICS. 


Book II. 


"Plate 1. 
Fig. 8. 


Plate 1. 
Fig. 9. 


the direction of, the three sides of the triangle GED : complete the parallelogram GEFD ; and make 
AD equal to, and in the direction of, the diagonal ED. 

If the body at D he acted upon by the forces AD, ED, equal and in opposite directions, it will be 
kept at rest. But the force ED (Prop. XVI.) is equivalent to the two forces DG, DF, that is, DG, GE; 
therefore the body acted upon by the three forces AD, DG, DF, that is, by three forces proportional to, 
and in the direction of, the sides of the triangle GED, will be at rest. 

Exp. Let three weights in the proportions of 3, 4, 5, be suspended by cords, which pass over pul¬ 
leys and meet in a point; if the directions of the cords be parallel to the sides of a triangle (drawn in a 
plane parallel to the plane of the cords) whose sides arc to each other as the weights, a ball at the 
point in which the cords meet will be kept at rest. 

Con. The body will be at rest if the three forces are proportional to the three sides of a triangle- 
drawn perpendicular to the direction of the forces; for such a triangle is similar to the former. Draw 
A g, C (/, and B e, perpendicular to the sides GE, GD, DE, forming a triangle g e d, which is equiangular 
to GED; hence, the sides about their equal angles being proportional, the forces which are proportional 
to the lines GE, GD, and DE, are also proportional to g c. gd , and d e. 

Scholium. A boy’s kite, as it rests in the air, is an instance of a body resting whilst three forces act 
upon it. For the kite is acted upon by the wind ; by its own weight; and by the string that holds it. 

PROP. XVIII. The force of oblique percussion is to that of direct or perpendicular 
action,, as the sine of the angle of incidence to radius. 

Let a body strike upon the plane AD, at the point D, in the direction BD : the line BD expressing 
the force of direct impulse may be resolved into two others, BC, BA, the one parallel, the other per¬ 
pendicular to the plane. Of these, the force BC, parallel to the plane, cannot affect it. The whole 
force upon the plane may therefore be expressed by BA. But BA is to BD as the sine of the angle 
of incidence BDA is to radius. 

Schol. If the surface to be struck be a curve, let AD be made tangent to the curve at D, and the 
proof will be the same. 

PROP. XIX. The force of oblique action produced by percussion is to that of di¬ 
rect action, as the cosine of the angle, comprehended between the direction of the force 
and that in which the body is to be moved, to radius. 

Let FD represent a force acting upon a body at D, and impelling it towards E; but let DM be the 
only way in which it is possible for the body to move. The force FD may be resolved into two forces 
FG, FH, or GD ; of which only the force GD impels it towards M. And, FD being radius, GD is the 
cosine of the angle FDG, or MDE, comprehended between the direction of the force, and that in which 
the body is to be moved. 


CHAPTER IV. 

Of Motion , as communicated by Percussion in Non-Elastic and Elastic Bodies. 

Def. II. Bodies are non-elastic, which, when one strikes another, do not rebound, 
but move together ofter the stroke. 

Cor. Hence their velocities after the stroke are equal. 

Def. III. Bodies are elastic, which have a certain spring, by which their parts, up¬ 
on being pressed inwards by percussion, return to their former state, throwing off the 
striking body with some degree of force ; when the elasticity is perfect, the body re¬ 
stores itself with a force equal to that with which it is compressed. 

Exp. The existence of this property is visible in a ball of wool, cotton, or sponge compressed. 

PROP. XX. When one non-elastic body in motion, strikes upon another at rest, or 
moving with less velocity in the same direction, the sum of their momenta remains the 
' same after the stroke as before. 



Chap. IV. 


OF ELASTIC BODIES. 


For (Prop. III. Cor. 1.) as much motion as the striking-body communicates, so much it loses; whence, 
if the motions of the bodies are in the same direction, whatever is added to the motion of the preceding 
body will be subducted from that which follows, and the sum will remain the same. 

PROP. XXI. When two non-elastic bodies, moving in opposite directions, strike 
upon each other, the sum of their momenta, after the stroke, will be equal to the dif¬ 
ference of their momenta before the stroke. 

For (from Prop. III. Cor. 1 .) that body which had the least motion will destroy a quantit} r equal to 
its own in the other; after which they will move together with the remainder, that is, the difference. 

Exp. Let two cylinders tilled with clay, A, B, be of equal weight, and suspended by cords from equal 
heights; let two other cylinders of the same kind, C, D, but in weight as 2 to 1 , be suspended from the 
same height. The heights from which they are let fall, in the arc formed by the motion of the cylin¬ 
der (from the nature of the pendulum, afterwards to be explained) will be the measure of their veloci¬ 
ty ; and (by Prop. XI.) their momenta will be as their velocities multiplied into their quantities of mat¬ 
ter ; whence the cases of the two preceding propositions may be established by the following experi¬ 
ments. N. B. Quantity of matter is expressed by </, velocity by u, and momentum by in. 

No. 1. Prop. XX. Case 1. Let the cylinder A fall from the height of 18 inches, upon the cylinder B 
at rest. The momentum of A before the stroke (by Prop. XI.) is 18 ; for the quantity of matter is 1, 
and the velociiy 18 ; whence q 1 x v 18 = in 18. After the stroke, the quantity of matter being (Def. II.) 
2, and the velocity of each cylinder 9, the momentum will be 18 ; q 2 X v 9 = m 18. 

No. 2 . Case 2. Let A fall from 18 inches, and B from 9, in the same direction; their momenta 
before the stroke are 18 -j- 9 = 27 ; after the stroke, the quantity of matter will be 2, and the velocity 
13A ; whence v 13-1- x q 2 = in 27. 

No. 3. Prop. XXL Case 1 . Let the equal cylinders A and B fall in opposite directions, from the 
height of 12 inches; the momenta being equal and opposite, the motion of both will be destro} r ed. 

No. 4. Case 2 . Let A fall from the height of 12 inches, and meet B falling in the opposite direc¬ 
tion from 6 inches; their velocity after the stroke being 3, and quantity of matter 2, the momentum 
will be 6 ; q 2 X v 3 = in 6 . 

No. 5. Prop. XX. Case 3 . Let the cylinder C, double of the cylinder D, fall from 12 inches on D 
at rest. Before the stroke, the quantity of matter in C is 2 and its velocity is 12 ; whence its momen¬ 
tum is 24 ; q 2 x v 12 == m 24. After the stroke, the * velocity will be 8 , and quantity of matter 3 ; 
w hence q 3x^8 = m 24. 

No. 6 . Case 4. Let C fall from 12 inches, and D from 6 inches in the same direction. Before the 
the stroke, the velocity of C is 12 , and quantity of matter 2 ; whence its momentum is 24; q 2 x v 12 
= m 24 ; and the velocity of D is G, and its quantity of matter 1 ; whence q 1 x v G = m 6 ; therefore 
the whole momentum is 30. After the stroke, the velocity of the whole is 10, and the quantity of mat¬ 
ter 3 ; whence q 3 X v 10 — m 30. 

No. 7. Prop. XXL Case 3. Let C fall from G inches, and D from 12 , in opposite directions, the 
quantity of matter in C being 2 , and its velocity G; and the quantity of matter in D being 1 , and its ve¬ 
locity 12 , their momenta will be equal, and being opposite, will destroy each other. C q 2 x v G = in 
12 ; D q 1 Xv 12 — in 12. 

No. 8 . Case 4. Let C fall from 3 inches, and D from 12 , in opposite directions: Before the stroke, 
the momentum of C is G; q 2 X v 3 — m G, and the momentum of D is 12; q 1 x v 12 *= m 12 ; 
w'hence the difference of their momenta is G. After the stroke, the velocity is 2, and quantity of matter 
3 ; whence the momentum is 6 ; q 3 x v 2 — m G. 

PROP. XXII. When one elastic body strikes upon another of the same kind, the 
one loses, and the other gains, twice as much momentum, as if the bodies had been 
void of elasticity. 

For since (by Def. III.) perfectly elastic bodies, on percussion, restore themselves with a force 
equal to that with which they are compressed, whatever momentum is gained by one body, or lost by 
the other, on percussion, from the law of re-action, the same must be gained, or lost, from the power 
of elasticity. 

Con. 1 . If one of the bodies, considered as non-elastic, would lose more than half its momentum, as elas¬ 
tic, it loses more than all, (hat is, acquires a negative momentum in a contrary direction. 

Exp. The following experiments may be made with ivory balls suspended by strings; they cos 
respond with the preceding experiments on non-elastic bodies. 

Let A and B be equal balls; and let C be a ball double of the ball D» 


14 


OF MECHANICS. 


Book. II. 


No. 1 . A, failing from 18 inches on B at rest, has 18 degrees of momentum before the stroke; 
therefore, after the stroke, supposing the balls non-elastic, the same momentum belonging to the two 
equal balls together, each has 9 degrees of momentum, and A has lost and B gained 9. This being 
doubled, A, as elastic, will lose 18, and B will gain 18 degrees of momentum: whence A will be at rest, 
and B will move with 18 degrees of momentum. 

No 2 . A, falling from 18 inches, and B from 9 in the same direction; as non-elastic after the stroke, 
each has 13£ momentum, or A has lost 44, and B gained 4^. As elastic, after the stroke, A loses 9, and 
B gains 9 ; therefore A rises to 9 inches, B to 18. 

3. A and B, falling in opposite directions from 12 inches, as non-elastic, would lose all their momen¬ 
tum ; as elastic, each loses 24 degrees of momentum; that is, gains 12 in the contrary direction. 

No. 4. A, falling from 12 inches, and B in the opposite direction from 6 , as non-elastic, the momen¬ 
tum of each, after the stroke, will be in the direction of A; whence A loses 9, and B loses 9, moving 
3 degrees in the contrary direction. As elastic, A loses 18, or has 6 in the contrary direction, and B 
loses 18, or gains 12 in the contrary direction. 

No. 5. C, double of D, falling from 12 inches on D at rest, the momentum of C, before the stroke, 
being 24, and of D nothing; as non-elastic, C, after the stroke, having its momentum 16, and moving 
with the velocity 8 , will have lost 4 degrees of velocity, and 8 of momentum: and D will have gained 
8 of each. As elastic, therefore, C will lose 8 degrees of velocity, or (Prop. XI.) 16 of momentum, and 
Ij will gain 16 of each; that is, C will move with 4 degrees of velocity, and D with 16. 

No. 6 . C, falling from 12 inches, and D from 6 in the same direction, before the stroke, the veloci¬ 
ty of C is 12, and its momentum 24; and the momentum of D 6 . After the stroke, as non-elastic, the 
momentum of C is 20 , because q 2Xv 10 = m 20 ; and the momentum of D is 10 , because q 1 10 

10; therefore C has lost 4 degrees of momentum, or 2 degrees of velocity, and D gained 4 of each. 
If, therefore, the gain or loss be doubled on account of the elasticity of C and D, C will lose 8 degrees 
of momentum, or 4 of velocity, and D will gain 8 of each ; that is, C will move with 8 degrees of veloc¬ 
ity, and D with 14. 

No. 7. C’, falling from 6 inches, and D from 12 , in opposite directions, their momenta, being equal, 
would destroy each other without elasticity : Therefore, being elastic, each will acquire the momen¬ 
tum of 12 in opposite directions; that is, D will l'eturn to 12 and C to 6 . 

No. 8 . C, falling from 3 inches, and D from 12 in opposite directions; since the momentum of C, 
before the stroke, is 6 , and of D 12 , as non-elastic bodies they would, after the stroke, move in the 
direction of D, with the velocity of 2 ; whence C would move in the direction contrary to its first mo¬ 
tion with 4 degrees of momentum, and lose 10 ; and D would lose 10 : Therefore, being elastic, C will 
lose 20 degrees of momentum, and also D 20 ; whence C will move in the contrary direction with 14 
degrees of momentum ; that is, will return to 7 ; and D will return to 8 . 

Cor. 1 . If the sum of two conspiring momenta, or the difference of two contrary momenta, be di¬ 
vided by the sum of the quantities of matter in both the moving bodies, the quotient will give the com¬ 
mon velocity after the stroke.* 

Schol. Let A and B be two spherical bodies, moving with their centres in the same line ; and let their 
velocities be a and b. The momentum of A, before the stroke, is A cr, and that of B is B b ; their sum, or 
their difference, is A a -f- B &, or A a — B 6 . Therefore (by Prop. XX. and XXI.) the momentum, after the 

stroke, is expressed b} f A a ± B 6 , and their common velocity by ——tt-- Hence the momentum 


of A, after the stroke, is 


AA a rt AB b 


and that of B is 


A -f B 
AB a =b BB b 


A + B 

Next, suppose the bodies perfectly elastic 
after the stroke, —i from its momentum, before the stroke, A a ; and the remainder, 


A + B 

Subtract the momentum of A considered as non-elastic, 


AB a AB b 


A -f- B 


A + B 


, will express the momentum in that case lost by A, and gained by B. 


mainder, 


AB a 


A -f B 


AB b , . ^ • r ^ , AA a 

, trom the momentum of A, as non-elastic, alter the stroke, 


Subtract this re- 
AB b 


add the same remainder to the momentum of B. after the stroke 


AB a rb BB b 


A A a ± 2 AB b — A B a 
A+~B 


A + B 

, will express the momentum of A, after the stroke, and the sum 
2 AB a d: BB b AB b 
A T~B 


A + B 
the difference, 


and 


* [This corollary belongs to the preceding proposition ; for it is true only of non-elastic bodies^ 















Chap. V. 


OF FALLING BODIES. 


15 


will express the momentum of B, after the stroke, supposing 1 them perfectly elastic. And 
Aa ± 2 B 6 — Ba 2 A a±BbzpAb 

-——---—, and - j -- jT - 1 -, will express their respective velocities. 

Cor. 2. If there be any number of elastic, equal, and spherical bodies, whose centres are placed in 
the same line, and the first body strikes upon the second in the direction of that line, all the bodies will 
be at rest except the last, which will move off with the velocity of the first. 

Exp. Several equal ivory balls, being so suspended as to have their centres in a right line, if the 
first be let fall upon the second, the last will fly off, to the height from which the first fell. 

Cor. 3. When the striking ball is less than the quiescent, there will be an increase of momentum. 

Exp. Let the ball D fall from 12 inches upon C, double of D, at rest. If they were non-elastic, they 
would proceed together, and, their velocity being the same, C, after the stroke, would have double the 
momentum of D ; that is, C would have 8 degrees, and D 4; whence D would have communicated more 
than half its momentum to C. The effect being doubled by the elasticity of the bodies, D communicates 
to C 16 degrees of momentum, and loses as much itself, or returns with 4 degrees af momentum in the 
contrary direction: while C moves forwards with 4 degrees more momentum than D had at the first. 
Thus the whole sum of momentum is increased from 12 to 20 degrees; but as much as the momentum 
is increased in the direction in which D first moved, so much is given to D in the contrary direction. 
In this manner may momentum be. continually increased by a series of bodies. 

Cor. 4. If a non-elastic body strike upon an immoveable obstacle, it will lose all its motion; an 
elastic body will return with a force equal to the stroke. 

Exp. Let a leaden ball,' and an ivory ball, strike upon any fixed plane. 


CHAPTER V. 

Of Motion, as produced by the Attraction of Gravitation. 

SECTION I. 

Of the Laics of Gravitation in Bodies falling without Obstruction. 

PROP. XXIII. The motion of a body, falling freely by the attraction of gravita¬ 
tion, is uniformly accelerated, or its velocity increases equally in equal times. 

A new impression being made upon the falling body, at every instant, by the continued action of the 
attraction of gravity, and the effect of the former (by Prop. I.) still remaining, the velocity must contin¬ 
ually increase. Suppose a single impulse of gravitation, in one instant, to give it one degree of velocity ; 
if, alter this, the force of gravitation were entirely suspended, the body would continue to move with 
that degree of velocity, withoufibeing accelerated or retarded. But, because the attraction of gravitation 
continues, it produces as great a velocity in the second instant as in the first; which being added to the 
first, makes the velocity in the second instant double of what it was in the first. In like man¬ 
ner, in the third instant, it will be tripled; quadrupled in the fourth ; and in ever}' instant, one degree 
of velocity will be added to that which the body had before; that is, the motion will be uniformly ac¬ 
celerated.* 

Cor. The velocities of falling bodies are as the times in which they are acquired. 

PROP. XXIV. The force of the attraction of gravitation acting upon any body is 
as its quantity of matter. 

Tor each particle of matter in any body being acted upon by gravitation, the greater number of 
particles are contained in any body, the greater force must be exerted upon it; that is, the force in¬ 
creases as the quantity of matter increases. 

Exp. Let two unequal balls, suspended by threads of the same length, be let fall at the same time from 
points equally distant from the lowest points of the arcs in which they move: The vibrations of each 

♦ All bodies descending in vacuo by gravity, whether great or small, dense or rare, are found to fall through 16.1 feet in 
one second, and to acquire a velocity in falling which would carry them uniformly through 32.2 feet in the next second, 
and an increase of velocity, equal to this, is found to be added to every succeeding seCond of time. 





16 


OF MECHANICS. 


Book II. 


will be performed in equal times, and consequently their velocities will be equal; wnence the momenta 
(Prop. XI.) will be as the quantities of matter; but (Prop. XIII.) the force producing motion, is as the 
quantity of motion, or momentum produced: Therefore the force of gravitation is as the quantity of 
matter; that is, as much greater force is exerted upon the larger body than upon the less, as its quantity 
of matter is greater than that of the less. 

Cor. 1 . The weight of anybody is as its quantity of matter; for weight is the degree offeree with 
which any body is acted upon by gravitation. 

Con. 2. If the attraction of gravitation were increased in any ratio, the weight of a given body 
would be increased in the same ratio. Substituting, therefore, W, Q, F, for the weight, quantity of 
matter, and force of gravity, respectively, and supposing them to be variable; W will be as Q. x F. 

PROP. XXV. The velocities of bodies falling from the same height, without re¬ 
sistance, are equal. 

If two bodies of different quantities of matter fall from the same height, the attracting force which 
acts upon the greater body r , will (Prop. XXTV.) exceed that which acts upon the less, as much as the 
greater body exceeds the less in quantity of matter; whence they must move with equal velocities. 

Exr. A guinea, and a feather, or other light body, in the exhausted receiver of an air-pump will fall 
through the same space in the same time. 

PROP. XXVI. The spaces described by falling bodies are as the squares of the 
times from the beginning of the fall, and also as the squares of the last acquired veloci¬ 
ties; or in the ratio compounded of the times and velocities. 

Plate i. In the triangle ABC, let AB express the time in which a body is falling, and BC the velocity which 

Fig. 10. has aC q U i r ed at the end of the fall; let AF, AD, he parts of the time AB ; and through F, D, draw FG, 

DE, parallel to BC. 

Because the triangles ABC, ADE, are similar, AB is to AD as BC to DE ; hut AB and AD, express 
times of descent, and BC expresses the velocity acquired in the time AB ; therefore since (Prop. XXIII. 
Cor.) the velocities are as the times, DE expresses the velocity acquired in the time AD. In like 
manner GF, any other right line parallel to BC, expresses the velocity acquired in the time AF. There¬ 
fore the sum of the lines which may be supposed drawn parallel (o CB in the triangle ADE ; that is, the 
whole triangle ADE, will represent the sum of the several velocities with which the falling body 
moves in the time in AD. For the same reason, the triangle ABC will represent the sum of the veloci¬ 
ties with which the falling body moves in the time AB. Since therefore it is manifest, that the space 
which a body passes through in any moment of time is as the velocity with which it moves at that 
moment; and consequently, that the spaces through which it passes in any times whatsoever, are as the 
sums of the velocities with which it moves in the several moments of those times; the spaces passed 
through in the times AD, AB, are to each other as the triangles ADE, ABC. But the triangle ADE 
(El. VI. 19.) is to the triangle ABC in the duplicate ratio of the homologous sides AD, AB, and also oi 
DE, BC; that is, the spaces are as the squares of the times, and also as the squares of the last acquired 
velocities; consequently the spaces described are in the compound ratio of the times and the velocities. 

Exr. Let there he two pendulums, one of which vibrates twice as fast as the other, a hall let fall 
from such a height above the ball of the shorter pendulum as to reach it in one vibration, must fall 
from four times this height, to reach the longer pendulum in one of its vibrations. 

Cor. 1 . Hence, if the forces are variable, the spaces described are as the forces and squares of the 
times; or as the squares of the velocities directly, and forces inversely. For by the Prop, (calling- S, 
V, and T, the space, velocitv, and time) S is as T x V, and (by Prop. A. p. 9.) V is as F x T .-. S is as 

V 2 V V V 2 

F x T 2 ; also, S is as ; for T is as -S is as V x -p or as —. 

r r r r 

Cor. 2. The times in which bodies fall from unequal heights, and their last acquired velocities, are 
as the square roots, or in the subduplicate ratio of their heights. Since TT is as S, T will be as ^/S; 
and since VV is as S, V will he as y/ S. 

Cor. 3. If the time of the fall of a body he divided into equal parts, the spaces through which it 
falls in each of these parts, taken separately, will be as the odd numbers 1, 3, 5, &c. The spaces being 
as the squares of the times or velocities, if the times be as the numbers 1, 2, 3, 4, the spaces will be as 
1, 4, 9, 1G ; whence, in the first time the space will be as 1, in the second time, the space passed over 
will be as 3, in the third, as 5, &c. 

Cor. 4. The space passed through during any portion of the time a body is falling, is always pro¬ 
portional to the difference of the squares of the velocities at the beginning and end of that portion 
of time. 


Plate 1. 
Fis;. 10. 


Chap. V. 


OF FALLING BODIES. 


I 


For (by the Prop.) AFG : A f g : : FG* : f g * ; therefore AFG — A f g oc/FUg : FG 3 — fg 2 : : 

AFG : FG 1 , and ADE : A de : : DE 3 : d e 2 ; therefore ADE — A de or dDEe : DE 2 — d e* : : ADE 
: DE* ; but AFG : FG 2 : : ADE : DE 2 ; therefore/FG g : FG 2 —fg 2 : : d DEe : DE 3 — d e 2 . 

Schoi.. Since S is as T 2 , and as in the first second of time a body freely descending by the force of 
gravity fills through 16.1 feet, we easily find the space described in any given number of seconds; for 
S = 16.1 x T*. Thus in 5" a body will fall through 402 feet; for 16.1 x 25 = 402. Again, the 
spaces fallen through in the 1st, 2d, 3d, seconds, are 16.1 ; 16.1 x 3; 16.1 x 5, respectively. 

PROP. XXVII. The space which a body passes over in any given time from the be¬ 
ginning of the fall, is half that which it would pass over in the same time, moving with 
the last acquired velocity. 

For the triangle ABC (by Prop. XXVI.) expresses the space passed over in the time AB when the Plate 1 
motion is uniformly accelerated ; the last acquired velocity is expressed by BC ; and the rectangle of Fig. 10. 
AB, BC, rightly expresses the space passed through in the time AB with the equable velocity BC ; since 
therefore the triangle ABC is half of the rectangle AB, BC, the proposition is manifest. 


PROP. XXVIII. The motion of a body thrown upwards is uniformly retarded by 
gravitation : the time of its rise will be equal to that in which a body "falling freely 
acquires the same degree of velocity with which it is thrown up ; and the height to 
which it will rise will be as the square of the time, or first velocity. 


The same force which accelerates a falling body, acting in an opposite direction upon one thrown 
upwards, must retard it: and, since the action of gravitation is uniform, in whatever time it generates 
■any velocity in a falling body, it must in the same time destroy the same velocity in a rising body : 
through whatever space the falling body must pass to acquire any velocity, the rising body must pass 
through the same to lose it; whatever ratio the spaces bear to the velocities and times in the one case, 
must take place in the other: the effect of gravitation in rising bodies being in all respects the reverse 
of its effect upon falling bodies. 

Schol. As the force of gravity near the surface of the earth is constant, and known by experiment, 
and as the spaces described by falling bodies vary as the squares of the times (T 2 ), or as the squares of 
the velocities (V 2 ); hence every thing relating to the descent of bodies, when accelerated by the force 
of gravity; and to their ascent, when they are retarded by that force, may be deduced from the forego¬ 
ing propositions. 

(I.) When a body falls by the force of gravity, the velocity acquired in any time, as T", is such as 
would carry it uniformly over 2 FT in 1"; where F = 16.1 feet. 

Exam. The velocity acquired in a falling body in 6" = 32.2 x 6, or such as would carry it uniformly 
through 193 feet in 1". 

V 2 _ V* 

(2.) The space fallen through to acquire the velocity V is — p. For S : F : : V 2 : orj 1 or S == —. 


Exam. If a body fall from rest till it acquire a velocity of 20 feet per second, the space fallen through 


is 


201 


4 X 16.1 


6.2 feet. 


From these three expressions, V = 2 FT ; S == — ; and S = FT* (Cor, 1. Prop. XXVI.); any one 


of the quantities S, T, V, being given, the other two may be found. 

Exam. 1. To find the time in which a body will fall 400 feet; and the velocity acquired. 

Since S =«= FT* .-. T = |-vr= Ir^r = 5" nearly, and V being etpial to 2 FT = 32.2 x 5" = 161 

^ r ^ 16.1 

feet «= velocity acquired. 

Exam. 2. If a body be projected perpendicularly downwards, with a velocity of 20 feet per second, 
to find the space described in 4''. 

The space described in 4" by the first velocity is 4 x 20, and the space fallen through by the action 
of gravity is 16.1 X 4 3 , therefore the whole space described is 337.6 feet. 

Exam. 3. To what height will a body rise in 3", which is projected perpendicularly upwards with a 
velocity of 100 feet per second ? 

The space described in 3''' by the first velocity is 300 feet, and the space through which the body 
would fall by gravity in 3''is 16.1 x 3 2 = 144.9 feet; therefore the h§iarht required is 300— 144.9 *= 
165.1 feet. 



18 


OF MECHANICS. 
SECTION II. 


Book. II. 


Of the Laws of Gravitation in Bodies falling down inclined Planes. 


Plate 1. 
Fig. 11. 


Def. IV. An inclined plane is a plane which makes an acute or obtuse angle with 
the plane of the horizon. 

PROP. XXIX. .The motion of a body, descending down an inclined plane, is uni¬ 
formly accelerated. 


In every part of the same plane, the accelerating force has the same ratio to the force of gravita¬ 
tion acting freely in a perpendicular direction, and is therefore (El. V. 9.) equally exerted in every in¬ 
stant of the descent; whence (as was shown concerning bodie s ailing freely, Prop. XXIII.) the motion 
must be uniformly accelerated. 

Cor. Hence, whatever has been demonstrated concerning the perpendicular descent of bodies, is 
equally applicable to their descent down inclined planes, the motion in both cases being uniformly accel¬ 
erated by the same power of gravitation. 


PROP. XXX. The force, with which a body descends by the attraction of gravita¬ 
tion down an inclined plane, is to that with which it would descend freely, as the ele¬ 
vation of the plane to its length ; or as the sine of the angle of inclination to radius. 


Let AB be the length of an inclined plane, and AC its elevation, or perpendicular height. If the 
force of gravitation with which any body descends perpendicularly be expressed by AC, and this force 
be resolved into two forces, AD, DC, by drawing CD perpendicular to AB ; because the force CD is 
destroyed by the reaction of the plane, the body descends down the inclined plane only with the force 
AD. And (El. VI. 8. Cor.) AD is to AC, as AC is to AB ; that is, the force of gravitation down the in¬ 
clined plpne is to the same force acting freely, as the elevation of the plane is to its length, or as the 
sine of the angle of inclination ABC is to the radius AB. 

Cor. 1. Hence, the force necessary to sustain a body on an inclined plane, is to the absolute weight 
of a body, as the elevation of the plane to its length, for the force requisite to sustain a body must be 
equal to that with which it endeavours to descend; which has been shown to be to that with which it 
would descend freely, as the elevation of the plane to its length. 

Cor. 2. If H be the height of an inclined plane, L its length, and the force of gravity be represented 


by unity; the accelerating force on the inclined plane is represented by 


H 


For by the Prop, the ac¬ 


celerating force is to the force of gravity (1) as H is to L .-. the accelerating force = 

Cor. 3. Hence -y- varies as the sine of the angle of inclination. 

L 


Cor. 4. If a body fall down an inclined plane, the velocity V generated in T" is such as would carry 

H J 

it uniformly over y x 2 FT feet in 1", where, as before, F is equal to 16.1. 


For (by Prop. A. p. 9.) the velocity varies as the force and time, i. e. as — x T, and the veloci- 

Li 

H 

ty generated by the force of gravity in one second is 2 F, therefore V = — x 2 F T. 

Li 

Ex. If L : H : : 2 : 1, a body falling down the plane will, at the end of 4", acquire a velocity of 
\ X 32.2 x 4 = 64.4 feet per second. 

TT 

Cor. 5. The space fallen through in T" from the state of rest,is— x FT a , for (Prop. XXVI.) the 

1 / k ' 

spaces described vary as the squares of the times. 

Ex. 1. If H = ~, the space through which a body falls in 5" is | x 16.1 x 25 = 2021 feet. 

Ex. 2. To find the time in which a body will descend 40 feet down this plane. Since S = ?XFT*. 

therefore T = secon *' 






Chap. V. 


OF INCLINED PLANES. 


19 


Cor. 6 , The space through which a body must fall, from a state of rest, to acquire a velocity V, is 
L V* V 2 

— X ypr* For (Cor. 1. Prop. XXVI.) S is as —, therefore the space through which the body falls by 

the force of gravity, is to the space through which it falls down the plane, as the square of the velocity 
directly, and as the force inversely in the former case, is to the same in the latter; and if F (16.1) be 


the space fallen through by gravity, 2 F is the velocity acquired in 1"; hence F : S : : 

L V 2 

and S = tt - X tft- 
H 4r 


2F| a 


_L_ 

IT 


X V' 2 , 


Ex. 1 . If L = 2 H, and a body fall from a state of rest till it has acquired a \«elocity of 40 feet per sec- 
2 40 2 

ond, the space described is — x , = 50 feet nearly. 

1 64.4 

Ex. 2 . If a body fall 40 feet from a state of rest down this plane, to find the velocity acquired. 
V 2 a 4FS X 7 = 64.4 x 40 x \ — 1288, and V = 35.8 feet per second. 


PROP. XXXI. The space described in any given time by a body descending down 
an inclined plane, is to the space through which it would fall perpendicularly in the 
same time, as the elevation of the plane to its length. 

Let AC represent the force with which a body would fall perpendicularly ; CD being drawn from C Plate 1 
perpendicular to AB; AD, as was shown (Prop. XXX.), will represent the force with which the body * ' 8 - 11 
descends down the inclined plane AB. And, since the spaces through which bodies fall in any given 
time must be as the forces which move them, the space through which the body will fall down the in¬ 
clined plane AB, is to that through which it will fall perpendicularly in the same time, as the force AD, 
to the force AC. But AD is to AC (El. VI. 8 . Cor.) as AC the elevation to AB the length of the plane ; 
therefore the space through which the body will fall in a given time down the inclined plane AB, will 
be to the space through which it would fall perpendicularly in the same time, as the elevation of the 
plane to its length. 

Cor. 1 . A body would fall down the inclined plane from A to D, in the same time in which it would 
fall perpendicularly from A to C. For, the spaces passed through in any given time are as AC to AB, 
that is, (El. VI. 8 . Cor.) as AD to AC ; consequently, if AC is the space passed through in any given 
time by the body falling freely, AD will be the space passed through in the same time, dow r n the in¬ 
clined plane AB. 

Cor. 2. Having the space through which a body falls in a perpendicular direction, we can easily 
find the space which a body will describe in the same time, on planes differently inclined, by letting 
fall perpendiculars, as CD, on those planes respectively. 


PROP. XXXII. The velocity, acquired in any given time by a body descending 
down an inclined plane, is to the velocity acquired in the same time by a body fall¬ 
ing freely, as the elevation of the plane to the length. 

In an uniformly accelerated motion, the velocities produced in equal times are as the forces which 
produce them ; but (by Prop. XXX.) the force with which a body descends down an inclined plane, is 
to that of its perpendicular descent, as the height of the plane to its length; therefore the velocities 
produced in equal times are in the same ratio. 

PROP. XXXIII. The time, in which a body moves down an inclined plane, is to 
that in which it would fall perpendicularly from the same height, as the length of the 
plane to its elevation. 

The square of the time in which AB is passed over, is to the square of the time in which AD is 
passed over (compare Prop. XXVI. with Prop. XXIX. Cor.) as AB to AD; that is, since AB, AC, AD, Plate 1 , 
(El. VI. 8 . Cor.) are continued proportionals, as the square of AB to the square of AC. Therefore the Fig- H 
times themselves are as the lines AB, AC ; that is, as the length of the plane to its elevation. 

Cor. Hence, if several inclined planes have equal altitudes, the times in which those planes are des¬ 
cribed by bodies falling down them, are as the lengths of the planes. For the time of the descent down p| a te 1 . 
AC is to the time of the fall down AB, as AC to AB; and the time of the tall down AB is to the time of Fig. 12 . 
the descent down AG, as AB to AG; therefore (El. V. 11 .) the time'of the descent from A to C is to the 
time of the descent from A to G, as AC to AG; that is, the times are as the lengths of the planes. 


2U 


OF MECHANICS. 


Book II. 


Plate 1. 
Fig. 11. 


Plate 1. 
Fig. 12. 


Plate 1. 
Fig. 12. 


Plate 1. 
I'ig. 13. 


Plate 1. 
Fig. 14. 


PROP. XXXIV. A body acquires the same velocity in falling down an inclined 
plane, which it would acquire by falling freely through the perpendicular elevation of 
the plane. 

The square of the velocity which a body acquires by falling to D, is (by Prop. XXVI. compared with 
Prop. XXIX. Cor.) to the square of the velocity it acquires by foiling to B, as the space AD is to the 
space AB, that is (El. VI. 8 . Cor.) as the square of AD is to the square of AC ; and consequently the 
velocity at D is to the velocity at B, as AD is to AC. But, because AD and AC (Prop. XXXI. Cor.) are 
passed over in the same time, the velocity acquired at D is (by Prop. XXXI1.) to that which is ac¬ 
quired at C, as AD to AC. Since then the velocity at D has the same ratio to the velocities at B, and 
at C, namely, the ratio of AD to AC, the velocities at B and C (El. V. 9.) are equal. 

Cor. 1 . Hence the velocities acquired by bodies falling down planes differently inclined are equal, 
where the heights of the planes are equal. The velocities acquired in foiling from A to C, and from A 
to G, are each equal to the velocity acquired in falling from A to B, and therefore equal to one another. 

Cor. 2 . Hence if bodies descend upon inclined planes whose heights are different, the velocities 
will be as the square roots of their heights. For (Fig. 8 and 9) the velocity in D is equal to that in A, 
and the velocity in D is equal to that in G. Therefore the velocity in D (Fig. 8 .) is to that in D (Fig. 
9.) as v'AB is tOv/FG (by Cor. Prop. XXVI.) 

PROP. XXXV. A body falls perpendicularly through the diameter, and obliquely 
through any chord of a circle, in the same time. 

In the circle ADB, let ABbe a diameter, and AD any chord ; draw BC a tangent to the circle at B ; 
produce AD to C, and join DB. Because ADB (El. III. 31.) is a right angle, a body (by Prop. XXXI. 
Cor.) will fall from A to D on the inclined plane in the same time in which it will foil from A to B per¬ 
pendicularly. In like manner let the chord AE be produced to G; and because AEB is a right angle, 
a body will fall from A to E on the inclined plane in the same time in which it would fall from A to B 

Cor. 1. Hence all the chords of a circle are described in equal times. 

Cor. 2. Hence also the velocities, and accelerating forces, will be as the lengths of the chords. 

PROP. XXXVI. If a body descends along several contiguous planes, the velocity 
which it acquires by the whole descent, provided it lose no motion in going from one to 
another, is the same which it would acquire if it fell from the same perpendicular 
height along one continued plane ; and this velocity will be the same with that which 
would be acquired by the perpendicular fall from the elevation of the planes. 

Let AB, BC, CD, be several contiguous planes; through the points A and D, draw HE, DF, parallel 
to the horizon, and produce the contiguous planes 'CB, CD, to G and E. By Prop. XXXIV. Cor. the 
same velocity is acquired at the point B, whether the body descends from A to B, or from G (o B. 
Therefore, the line BC being the same in both cases, the velocity acquired at C must be the same, 
whether the body descends through AB, BC, or along GC. In like manner, it will have the same ve¬ 
locity at D, whether it falls through AB, BC, CD, or along ED ; that is, (by Prop. XXXIV.) its velocity 
will be equal to the velocity acquired by the perpendicular fall from H to D. 

Cor> Hence, if a body descend along any arc of a circle, or any other curve, the velocity acquired 
at the end of the descent is equal to the velocity acquired by falling down the perpendicular height of 
the arc; for such a curve may be considered as consisting of indefinitely small right lines, representing 
contiguous inclined planes. 

Schol. The velocity of a body, passing from one inclined plane to another, is diminished in the ra¬ 
tio of radius, to the cosine of the angle between the directions of the planes. Let BC, or B m (Fig. 20.) 
represent the velocity acquired at B, and resolve BC into Bn and C n, by letting foil the perpendicular 
C n ; mn will be the velocity lost; therefore the velocity at B is to the velocity diminished by passing 
from AB to BD as BC to B n, or as radius to the cosine of the angle between the directions of the planes. 

PROP. XXXVII. If two bodies fall down two or more planes equally inclined, and 
proportional, the times of falling down these planes will be as the square roots of their 
lengths. 

Let the inclined planes be AB, BC, DE, EF; let AG, DH, be lines drawn parallel to the horizon-; 
let AB, DE, be equally inclined to the plane of the horizon, and also BC, EF; let AB be to DE as AG 
to DH and as BC to EF, and draw GB, HE. 


Chap. V. 


OF PENDULUMS. 


21 


Because ABG, DEH, are similar triangles, AB is to DE (El. VI. 4.) as BG to EH, and v/AB 
to v/DE as ^/BG to ; also AB is to DE as BG -f BC is to HE + EF, and ^/AB to ,/DE as 

\/BG + BCj or v/GC, is o\/HE + EF, or ^/IIF. 

And since (by construction) AB is to DE as BC to EF, AB is to DE as AB -(- BC is to DE -f- EF, 
and ^/AB to v/DE, as n/AB + BC to \/DE -f EF. But AB, DE, being planes equally inclined, the 
accelerating force of g ra vitation will be the same upon each, and the bodies descending upon them may 
be considered as falling down different parts of the same plane. Hence, (Prop. XXVI. Cor. 2. and 
XXIX. Cor.) the time of descent along AB is to that along DE, as v/AB to v'DE ; and the time of des¬ 
cent along GC is to that along HF, as ,/GC to ,/HF ; that is, as ,/AB to v/DE. Again, the time of 
descent along GB is to that along HE as y/BG ,/EH; that is, as ,/AB to y/DE. Since, therefore, the 
time of descent along the whole plane GC is to that along the whole plane HF, as y/AB to y/DE, and that 
of the part GB is also to that of the part HE, as y/AB to y/DE, the time of descent along the remainder 
BC is to that-along the remainder EF (El. V. 19.) as y/AB to y/DE. Consequently, the time of descent 
down BA + BC is to that down DE -f- EF, as y/AB to ./'DE; that is, as \/AB -f- BC to v^DE -f- EF. 

Cor. Hence, if bodies descend through arcs of circles, the times of describing similar arcs will be 
as the square roots of the arcs. For such similar arcs may be considered as composed of an equal num¬ 
ber of proportional sides, or planes, having the same inclination to each other, and their elevations 
equal; whence, by this proposition, the times of descent will be as the square roots of the lengths of 
the arcs. 

PROP. XXXVIIL If a body be thrown up along an inclined plane, or the arc of a 
curve, it will, in the same time, rise to the same height, from which, with equal force, 
it would have descended ; and any velocity will be lost in the same time in which it 
would, in descending, have been acquired. 

For the force of gravitation has, in every respect, the same efficacy to retard the motion of bodies 
ascending, as to accelerate them descending on an inclined plane or curve. 

SECTION III. 

Of the Pendulum and Cycloid. 

Def. V. A pendulum is a heavy body, hanging by a cord or wire, and moveable 
with it upon a centre. 

PROP. XXXIX. The vibrations of a pendulum are produced by the force of 
gravitation. 

Let the ball A, suspended from the centre B by the cord BA, be drawn up to C and let fall from Plate i. 
thence: it will descend by the force of gravitation to A, from whence (being prevented from falling Fig. 15. 
farther by the cord) it will proceed (by Prop. XXXVI. Cor.) with a velocity equal to that which it 
would have acquired in falling perpendicularly from E to A, which will carry it on the opposite side 
to the height from which it fell. Being brought back again towards A by the force of gravitation, it 
will acquire a new velocity which will carry it towards C ; and in this manner it will vibrate by tire 
force of gravitation, till the resistance of the air, and the friction of the string, stop its motion. 

PROP. XL. The same pendulum, vibrating in small unequal arcs, performs its 
vibrations nearly in equal times. 

In the circle CGA, the small arcs CA, EA, will differ little from their respective chords in length or de- p] a t e i 
clivity. But (by Prop. XXXVI. Cor.) the times in which the chords are passed over are equal; there- Fig. 16 
fore the times of describing the arcs CA, EA, and also (by Prop. XXXVIII.) of describing their doubles and 1 " 
CAD, EAF, will be nearly equal. 

Exp. Two equal pendulums, vibrating in small, but unequal arcs, will, for a long time, keep pace in 
their vibrations. 

PROP. XLI. If a pendulum vibrate through small arcs of circles of different 
lengths, the velocity it acquires at the Iqwest point, is as the chord o£ the arc which it 
describes in its descent. 








22 


OF MECHANICS. 


Book II. 


Plate 1. 
Fig. lf> 
and IT. 


Fig. 17. 


Plate 1. 
Fig. 18. 


Fig. 19. 


Let BA be the pendulum, and CAD, EAF, the arcs through which it vibrates; and draw the hori¬ 
zontal lines EK, CPI. The velocity acquired in falling from H to A is (by Prop. XXXVI. Cor.) to that 
acquired in falling from G to A, as yHA to ,/GA ; that is, (by El. VI. 8. Cor.) as CA to GA. For the 
same reason, the velocity acquired in falling from G to A, is to that acquired in falling from K to A, as 
GA to EA. Consequently, ex cequali , the velocity acquired in falling from H to A is to that acquired in 
falling from K to A, as CA to EA. But (by Prop. XXXVI. Cor.) the velocity acquired in falling from If to A 
is equal to that from C to A ; and the velocity acquired in fallingfrom K to A is equal to that from E to A. 
Therefore the velocity acquired in descending through the arc CA is to that through EA, as the chord CA is 
to the chord EA ; and the same may be shown concerning the remaining half of the vibrations, AF, AD,. 

Cor. Hence the lengths of the chords of arcs, through which pendulums move, arc measures of ve¬ 
locity. 

PROP. XLII. The time of the descent and ascent of a pendulum, supposing it to 
vibrate in the chord of a circle, is equal to the time in which a body, falling freely, 
would descend through eight times the length of the pendulum. 

For the time of the descent of a body upon the chord is (by Prop. XXXV.) equal to that of the fall 
through the diameter of the circle, which is twice the length of the pendulum ; but in double that time, 
that is, in the descent and ascent, or whole vibration, the body would fall (by Prop. XXVII.) through 
four times the space, that is, through eight times the length of the pendulum. 

PROP. XLIII. The times in which pendulums of different lengths perform their 
vibrations, are as the square roots of their lengths. 

Let the two pendulums, AB, CD, be of different lengths. The time in which the first, AB, vibrates 
through a chord, is equal to that in which a body (Prop. XXXV.) would fall freely through twice AB, the 
diameter of the circle of which AB is radius. In like manner, the time in which CD vibrates is equal 
to that in which a body would fall through twice CD. But the times in which a body would fall through 
these different spaces are (Prop. XXVI. Cor. 2.) as the square roots of the spaces ; that is, as the square 
roots of AB and CD, the lengths of the pendulums ; therefore the vibrations are in the same ratio. 

Cor. The times in which pendulums of unequal lengths vibrate, are as the square roots of the simi¬ 
lar arcs through which they move. Let BA, BC, be pendulums of different lengths, vibrating in the 
similar arcs FG, DE. Since the times of vibration are as the square roots of the lengths BA, BC, and 
similar arcs are as the diameters, the times of vibration are as the square roots of the arcs, FA, DC, or of 
their doubles, FG, DE. 

Exr. Two pendulums, the lengths of which are as 1 to 4, will perform their vibrations in times 
as 1 to 2 ; that is, the shorter pendulum will make two vibrations, whilst the longer makes one ; for 
T : t : : -y/L : v' /. 

PROP. XLIV. The squares of the times in which a pendulum of a given length 
performs its vibrations, are inversely as the accelerating forces, or gravities. 

B}f Prop. XXVI. where the accelerating force is given, the space described is as the square of the 
time in which it is described. And since, in any given moving body, the velocity is as the accelerating- 
force (Prop. A. p. 9.) where the square of the time, or the time itself, is given, (by Prop. II.) the space 
described will be as the accelerating force. Consequently, where neither the accelerating force, nor 
the square of the time, is given, the space described will be in the ratio compounded of both. If then 
the space described be called S, the accelerating force A, and the square of the time T 3 , S will be as 
T 2 A S 

TVA ; whence ———, or T 2 , is as But, when the spaces are equal, S is a given quantity ; whence 

g 

(since fractions, whose numerators are given, are inversely as their denominators,) L-is inversely as A. 

s 

But T 1 is as-L; therefore where S is given, T* is inversely as A; that is, where the spaces de¬ 
scribed are equal, the squares of the times in which they are described are inversely as the accelerat¬ 
ing forces. And if the squares of the times of falling bodies are inversely as their accelerating forces, 
the squares of the times, in which pendulums vibrate, are in the same ratio, on account of the constant 
equality between the time of vibration and that of the descent through eight times the length of the 
pendulum, by Prop. XLII. 

Cor. 1 Hence, if the same pendulum, at different parts of the earth, perform its vibrations in differ* 
ent times, the forces of gravitation will, in those places, be inversely as the squares of those times. 



Chap. V. 


OF PENDULUMS. 


23 


Cor. 2. If the vibrations of pendulums of unequal lengths be performed in the same time, the accele¬ 
rating forces will be as their lengths. For (by Prop. XLIII. and XLIV.) T 2 : t 3 : : : —; therefore, 

A Cl 

when T = t, A : a : : L : /. Hence, as it is known by experiment, that the lengths of pendulums that 
vibrate seconds are diminished in approaching the equator, the force of gravity must also decrease. 

Ex. At the equator a pendulum vibrating seconds is -Aj-th of an inch shorter than such a pendulum in 
the latitude of London, and the length of this pendulum in London is 39.2 inches ; therefoi*e gravity un¬ 
der the equator is to gravity here as 391 is to 392. 

LEMMA I. 

If, from X as the centre, with any distance XA> a quadrant of a circle ADB be des - Plate 2. 
embed, and in the right line AX a body descends with such force, that its velocity in any 1 
points M, N ? &c. shall be always as MD, NP, c . the sines of the arcs AD, AP : the 
time in which the body will descend from A to X, will be equal to the time in which it 
would describe the whole arc ADB, with the uniform velocity, expressed by XB, acquired 
by the falling body when it arrives at X ; also, the time of the fall through any space 
AM, will he to the time of the fall through any other space AO, as the arc AD to the 
arc AQ,; and the force, with which the body is accelerated in any place M, will be<is 
MX, the distance of that place from the centre. 

Let DP be a part of the circumference taken indefinitely small, and therefore not assignably differing 
from a right line ; join DX ; and draw DL perpendicular to NP. Because the triangles MDX, LDP, 
are similar (having each a right angle, and, the angles MDX, LDP, whose common complement is LDX, 
equal) MD will be to DX as LD or MN to DP. But, by the hypothesis, MD is as the velocity of the 
descending body at the point M, that is, as the velocity with which the indefinitely small line MN is 
described; and XD is as the velocity last acquired by the falling body at X, that is, as the uniform ve¬ 
locity with which the arc DP is described. The velocity therefore of the body descending through the 
indefinitely small line MN will be to the velocity of the body moving along the arc DP, as MN to DP. 
Wherefore, since the velocities are proportional to the spaces passed over, the times wherein those 
spaces MN, DP, are described, will be equal. After the same manner it may be proved, that any other 
indefinitely small portion of the circumference, PQ, may be described with the velocity XB, in the same 
time in which the corresponding line NO will be described with the corresponding velocity NP ; and 
consequently, by composition, the falling body will descend through all the indefinitely small portions of 
the perpendicular AX, that is, through the whole line, in the time in which all the corresponding parts 
of the circumference, that is, the whole quadrant ABD, is described with a uniform velocity as XB. 

Moreover, the time in which the falling body descends from A to M, is equal to the time in which 
the arc AD is passed over; and the time in which it descends from A to O is equal to the time in which 
the arc AQ, is described ; but the time in which the arc AD is passed over, is to that in which the arc 
AQ is passed over, (since they are both described with the same velocity) as the arc AD to the arc AQ ; 
therefore the time of descent from A to M, will be to the time of descent from A to O, as the arc AD 
to the arc AQ; and consequently, by division, the time of descent through AM will be to the time of 
descent through MO, as the arc AD to the arc DQ. 

Lastly, let the arcs DP, PQ, be equal; join XP, and from P let fall PS perpendicular to OQ. The 
time of descent through MN will be equal to that through NO ; and, since the triangles LDP, MDX, 
are similar, and also SPQ, NPX; LP will be to DP or PQ, as MX to XD or XP ; also PQ is to SQ as 
XP to XN ; and consequently (El. V. 11.) LP will be to SQ as XM to XN. But LP is as the increment 
of the velocity acquired while the body is passing over MN, and SQ is as the increment of the veloci¬ 
ty acquired in passing over in an equal time the indefinitely small line NO ; and the forces with which 
the body is accelerated at M and N, are as the increments of thq velocities generated in equal times; 
the accelerating forces at M and N, will therefore be as the lines LP, SQ ; that is, the force with which 
the body is impelled at M is to that at N, as the distance XM to the distance XN, or the accelera¬ 
ting forces are as their distances from the centre. 

Cor. Hence, conversely, if a body, descending from A to X, is impelled by a force which is as its distance 
from the centre X, and the force at the beginning of the motion is expressed by the right line CE (the arc 
AE being taken indefinitely small), the velocities of the same body in any places M, O, will be expressed 
by the sines MD, OQ; and the times by the arcs AD, AQ; and the increments of the velocities, or, if 
'■he ai’cs increase equally, the accelerating forces, will be expressed by the increments of the sines - 


24 


OF MECHANICS. 


Book II. 


Plate 2. 
Fig. 2. 


Plate 2. 
Fig. 3. 


LEM. II. If a body , moving along the line AX, be hnpelled by forces proportional 
to its distance from the point X ; from whatever height it falls, it will arrive at the point 
X in the same time ; and this time will be to the time in which it would move over the line 
AX with the velocity which it acquires by falling from A to X, as half the circumference 
of a circle to its diameter. 

Let two bodies be let fall from the points A and P at the same time; and let them be impelled by 
forces proportional to their distances from the point X ; these bodies will come to X at the same time. 
From X as a centre, with the radii XA and XP, describe the two quadrants AB, PQ; and let the force 
by which the body A is impelled, or, which is the same thing, its velocity at the beginning of mo¬ 
tion, be represented by RS, the sine of the indefinitely small arc AS. It is manifest (from the Cor. of 
the preceding lemma) that its velocity, after the fall to X, will be properly expressed by XB. But, by 
hypothesis, the force by which the body at A is accelerated, is to that by which the body at P is accelera¬ 
ted, as AX is to PX, that is, (since the arcs AS and PN are similar) as RS to MX. As therefore RS 
expresses the first velocity of the body moving from A, MN will express the first velocity of the body 
moving from P; and consequently (by the Cor. to the last lemma) XQ will express the velocity of the 
bod}' moving from P, when it arrives at X. Farther, the time of the fall from A to X (by Lemma I.) is 
equal to the time in which the arc AB would be described with a velocity as XB ; and the time of the 
fall from P to X is equal to the time in which the arc PQ, would be described with a velocity as XQ. 
But (because the line XQ is to the line XB as the arc PQ to the arc AB, and the spaces passed over 
are proportional to the velocities) the time in which the arc AB is described with the velocity XB is 
equal to the time in which the arc PQ is described with the velocity XQ. Wherefore the time of the 
fall from A to X will be equal to the time of the fall from P to X. 

Again, since (by Lem. I.) the time in which a body would fall from A to X is equal to the time in 
which it would move over the arc AB, with its last acquired velocity at X; and since it is evident, 
that the time in which a body would move over the arc AB with the velocity at X is to the time in 
which it would move over AX with the same velocity, as AB is to AX ; the time in which a body 
would fall from A to X is to the time in which it would move over AX with the last acquired velocity 
as AB to AX. But AB is to AX, as twice AB to twice AX ; that is, as half the circumference of a cir¬ 
cle is to its diameter. Therefore the time in w'hich a body would fall from A to X is to the time in 
which it would move over AX with its last acquired velocity, as half the circumference of a circle is to 
its diameter. 

* 

Def. VI. If a circle, as FCH, be rolled along the line AB, till it has turned once 
round ; the point C in its circumference, which at first touched the line at A, will de¬ 
scribe the curve line ACXB, which curve is called a Cycloid. The right line AB is 
its base; the middle point X is its vertex; a perpendicular, as XD, let fall from 
thence to the base, is its axis; and the circle FCH, or any other, as XGD, equal 
thereto, is called the generating circle. 

LEM. III. If on XD, the axis of the cycloid , as a diameter , the generating circle 
XGD be described; and if from a point in the cycloid , as C, the line CIK be drawn par¬ 
allel to the base , the portion of it CG will be equal to the arc GX. 

Because‘the generating circles FCH, DGX, are equal (the diameter HF being drawn), KG is equal 
to Cl; w hence, adding G1 to both, KI will be equal to CG ; and KI, by construction, is equal to DF ; 
therefore CG is equal to FD. By the description of the cycloid, the arc CF is equal to the line AF; 
and by the construction the arc CF is equal to DG ; therefore AF is equal to DG; but, by the descrip¬ 
tion of the cycloid, AFD is equal to DGX; consequently, FD is equal to GX ; and CG was proved to be 
equal to FD ; therefore CG is equal to GX. 

LEM. IV. A tangent to the cycloid at the point C is parallel to GX, a chord of the 
circle DGX. 

Draw ck , parallel to the base and indefinitely near to CK, meeting the cyclbid in c, the axis in fc, 
and the circle in g. Let C u and G n, parallel to the axis, meet ck in u and «, and from T, the centre 
of the circle XGDM, draw' the radius TG. Since eg is equal (Lem. III.) to g X, gk being added to 
both, c k will be equal to X g + g k ; therefore c ??, the excess of c k above CK, is equal to G g -f- g n, 
the excess of Xg -f- g k above XG -j- GK. And, if we suppose c k to approach towards CK, as Gg and * n 
vanish^ the triangles Ggn and GTK become similar; for the angle gGnisthen equal to the «ngle 


Chap. V. 


OF THE CYCLOID. 


23 


TGK, since both have the the same angle »GT, or its alternate GTK, as their complement. Whence 
Gg is to g n as TG to TK, and (El. V. 18.) Gg -f gn to gn, as TG -f TK or DK to TK; hut G n is 
to g n as GK to TK; therefore Gg + ^ n is to G n as DK is to GK, that is, (El. VI. 8.) as GK to XK. 
And consequently cm (shown to be equal to Gg -f- gn) is to Gn, or Cm, as GK to XIv; and if the chord 
C c be drawn, the triangles Cue , XKG, will be similar; so that the chord Cc (as the points C and c 
coincide) becomes parallel to XG ; therefore the tangent of the cycloid at C is parallel to XG. 

LEM. V. If from a point of the cycloid, as L, the line LMK be drawn parallel to 
the base AB, the arc XL of the cycloid, will be double of XM, the chord of the circle 
corresponding thereto . 

Draw a line S h parallel and indefinitely near to LK crossing the circle in R, and the chord XM pro¬ 
duced, in P ; join the points X and ll ; on MP let fall the perpendicular RO ; and draw MN, XN, tan¬ 
gents to the circle at M and X. Then will the lines XN and k S, being each perpendicular to the diam¬ 
eter DX, be parallel; and the triangles MNX, MPR, having their angles at M vertical, and at P and X 
alternate, will be similar. But the tangents NX and NM are equal; (El. III. 36.) whence the lines PR 
and RM are also equal; the triangle RMP is therefore isosceles ; and RO being perpendicular to its base 
MP, MO (El. I. 26.) is equal to OP ; whence MP is equal to twice MO. The indefinitely small arc LS 
of the cycloid will not assignedly differ from a portion of a tangent drawn through the point L. LS may 
therefore (Lem. IV.) be said to be parallel to MP, and consequently (from the parallelism of ML and PS) 
equal to it; it is therefore equal also to twice MO. But LS is the difference between the cycloidal arcs 
XL and XS; and MO is the diffei'ence between the chords XM and XR ; for since XO and XR are in¬ 
definitely near to each other, RO, which is perpendicular to one of them, may be considered as perpen¬ 
dicular to both ; the indefinitely small difference therefore between any two arcs of the cycloid is twice 
that which is between the two corresponding chords of the circle ; and the same is true when the mag¬ 
nitude of the difference is assignable, because such difference is compounded of indefinitely small parts. 
Now, any arc whatsoever may be considered as a difference between two arcs, and consequently any 
arc, as XL, is double of the corresponding chord XM. 

Cor. Since when the arc XL becomes XB, the corresponding chord XM becomes XD, the diameter 
of the circle DMX; it is obvious, that the semicycloid BX, or AX, is equal to twice DX, the diameter 
of the generating circle DMX. 

LEM. VI. If a body descend in a cycloid, the force of gravity, so far as it acts 
upon the body in causing it to descend along the cycloid, will be proportional to the dis¬ 
tance of the body from the lowest point of the cycloid. 

Let the cycloid be AXB, whose base is AB, and its axis DX ; on which last, as a diameter, describe 
the generating circle DQX ; draw the chords OX and QX; through the points O and Q, and parallel to 
the base AB, draw the lines LS and MR; draw also the tangents LV and MY. Then because (by Lem. 
IV.) the tangent LV is parallel to OX, and the tangent MY parallel to QX, it is obvious that gravity ex¬ 
erts the same power upon a body descending in the cycloid at L, (because it then descends in the tangent 
LV) as it would do upon the same body descending along the chord OX; and, for the like reason, it 
exerts the same force upon it when it comes to M, that it would do if it were descending along QX; 
but (from Prop. XXXV.) the power or force of gravity upon bodies descending along the chords OX 
and QX, are as the lengths of those chords; that is, by Lem. V. (halves being proportional to their 
wholes)-as the length of the cycloidal arcs LX and MX. The force therefore of gravity upon a body 
descending in the cycloidal at the point L is to its force upon the same when at M (as may be said of 
any other corresponding points) as the space or distance it has to move over in the former case, before 
it reaches the lowest point X, to that which it has to pass over in the latter, before it arrives at the 
same point. 

PROP. XLV. If a pendulum be made to vibrate in a cycloid, all its vibrations, 
however unequal in length, will be performed in equal times. 

The force of gravity, (by Lem. VI.) so far as it causes a body to descend in a cycloid, is proportional 
to the distance of that body from the lowest point. Imagine then that body to be a pendulum vibrating 
in the cycloid, and from whatever point it sets out, it will (by Lem. II.) come to the lowest point in the 
same time; and consequently, since the same may be easily inferred in its ascending from that point, all 
its vibrations, be they large or small, will be performed in the same time. 

Schol. This proposition is demonstrated only on the supposition that the whole mass of the pendulum 
is concentrated in a point, for it cannot otherwise take phfee, because as the string varies in its length,' 1 

4 


Plate 2. 
Fig. 3 


Plate 2. 
Fig. 4. 


26 


Plate 2. 
rig. 4. 


Plate 1. 
Fig. IS. 


OF MECHANICS. Book II. 

the centre of oscillation of a body will vary. On this account, therefore, pendulums vibrating in circu¬ 
lar arcs are now always used, for the same arcs will be always described in the same time. 

PROP. XLVI. To make a pendulum vibrate in a given cycloid. 

Let AXB be the given cycloid ; its base AB, its axis DX, and its generating circle DQX, as before ; 
produce XjD to C, till DC is equal to DX ; through C draw the line EF parallel to AB, and take CEand 
CF, each equal to AD or DB ; and on the line CE as a base, and with the generating circle AGE 
equal to DQX, describe the semicycloid CTA, whose vertex will therefore touch the base ot the given 
cycloid in A. And on the line CF also as abase, describe an equal semicycloid CB. Let the semicycloids 
CA, CB, represent thin plates of metal bent to their figure, and on the point C, hang the pendulum CTP by 
a llexible line equal in length to the line CX. The upper part of its string (as CT, in its present situa¬ 
tion in the figure) as it vibrates, will then apply itself to the cycloidal cheeks CA and CB, and a bail at 
P will oscillate in the given cycloid AXB. 

Draw TG and PH each parallel to the base AB, and draw AG and DH. Then (Lem. V. Cor.) AC 
is equal to twice AE ; and by construction, twice DC ; that is, twice AE is equal to CX; therefore AC 
is equal to CX. Also, by construction, CTP is equal to CX; that is, to ATC ; whence, taking away 
CT, AT is equal to TP. By Lem. IV. GA is parallel to TP; and, by construction, AK is parallel to 
GT; therefore GA is equal to TK, and GT to AK ; but (Lem. V.) GA is halfTA; therefore TK is 
equal to halfTA ; since therefore it lias been proved that TA is equal to TP, TK is equal to half TP; 
that is, to KP. Hence it is manifest, that the parallel lines GT, PH, are equally distant from AD, the arc 
GA equal to the arc DH, the chords GA and D11 parallel, and GE equal to liX. And because GA has 
been shown to be parallel to TK, and also to DH, KP and DH are parallel; whence KD is equal to PII.. 
But (Lem. III.) GT, that is, AK is equal to the arc AG ; and by the description of the semicycloid CTA, 
AKD is equal to AGE; therefore KD is equal to EG ; that is, PPI is equal to HX. And (by Lem. Ill.) 
if PII be equal to IIX, P is a point in the cycloid AXB. The ball of the pendulum therefore, being at 
that point, is in the given cycloid. 

Schol. 1 . It is easy to conceive, that in a pendulum there must be some one point, on each side of 
w hich the momenta of the several parts of the pendulum will be equal, or in which the whole gravity of 
the pendulum might be collected without altering the time of its vibrations. This poiht, which is called 
the centre of oscillation , is ditferent from the centre of gravity ; for if a plane perpendicular to the 
string of the pendulum AB, he conceived to pass through the centre of the ball B, bisecting it; the ve¬ 
locity of the lower half, and consequently its momentum, will, in vibration, be greater than that of 
the upper half; consequently the centre of oscillation must be farther from A than the centre of gravi¬ 
ty is; and a plane passing through the centre of oscillation will divide the ball into two unequal parts, 
so that the greater quantity of matter above it shall compensate foi' the greater velocity below it, and 
the momenta on each side be equal. If the pendulum be an inflexible rod, every where of equal 
size, it is found, that the distance of the centre of oscillation from the point of suspension is two thirds 
of the length of the rod. 

If, whilst a pendulum is in motion, it meets with an obstacle at its centre of oscillation sufficient to 
stop it, the whole motion of the pendulum will cease at once, without any jarring ; for the obstacle re¬ 
sists equal momenta above and below this point; which is therefore also called the centre of percussion. 

Schol. 2. The vibrations of pendulums are subjected to many irregularities, for which no effectual 
remedy has yet been devised. These are owing partly to the variable density and temperature of the 
air, partly to the rigidity and friction of the rod by which they are suspended, and principally to the di¬ 
latation and contraction of the materials of which they are formed. The metalline rods of pendulums 
are expanded by heat, and contracted by cold ; therefore clocks will go slower in summer and faster in 
winter. The common remedy for this inconvenience is the raising or lowering of the bob of the pendulum 
(by means of a screw) as the occasion may require. By the last scholium it appears, that a pen¬ 
dulum consisting of a tube of glass or metal, every where uniform, filled with quicksilver, and 58.8 inches 
long, will vibrate seconds ; for | of 58.8 is equal to 39.2. Such a pendulum will be expanded and con¬ 
tracted at the same time; for when the tube is extended by heat, the mercury will also he expanded, 
and by rising in the tube, will raise the centre of oscillation* so that its distance from the point of sus¬ 
pension will he diminished, and the vibrations of the pendulum, which would have been rendered slower 
by the expansion of the tube, will become quicker by the expansion of the mercury; and, by adjusting 
the tube and mercury in such a manner, that these contrary effects may be the same, a clock with such 
a pendulum would admit of little or no variation for a long time. Phil. Trans. No. 392, p. 40. 


Chap. V. 


OF THE CENTRE OF GRAVITY. 


27 


SECTION IV. 

Of the Centre of Gravity. 

PROP. XLVII. In every body there is a centre of gravity , or a point about which 
all its parts balance each other. 

Let AB be an inflexible rod, throughout uniform and of the same density; let it be supported at the Plate 2. 
point C, equally distant from its extreme points A and B, by the prop C. Let A and B be indefinitely small 
and equal portions of the rod AB. These portions, A and B, tend towards the centre of the earth with 
equal forces of gravitation. They would likewise, without obstruction, move with equal velocities ; p- lfr $ 

for if the rod AB be moved on its prop till it come into the position DE, the velocities of the parts A, c 

B, or F, will be as the spaces over which they pass in the same time ; that is, as the arcs AD, EB, or 
FG, which arcs are as their respective circumferences, or as their diameters or radii ; whence the 
velocity of the part B is to the velocity of the part A, or F, as AC, or FC. And the quantities of matter 
in A and B are by supposition equal. Therefore, if the parts A and B were in motion, they would have 
equal momenta; that is, the efforts which A and B make to descend towards the earth are equal. But 
these efforts counteract each other; for, whilst the portion A endeavours, with a certain force, to draw 
down one arm of the rod, the other portion B endeavours with the same force to draw down the other 
arm; that is, since the rod is inflexible, to raise the portion A. Therefore the portion A is acted upon 
by two equal forces in contrary directions, and consequently must be at rest. For the same reason, the 
portion B will be at rest. And the same may be shown concerning any other equal portions, at equal 
distances from C, in the rod AB. Therefore the rod will be at rest; that is, the parts on each side of 
the point will balance each other, and C will be the centre of gravity. 

If the rod were placed oblique to the prop C, indefinitely small and equal parts being taken, as be¬ 
fore, at equal distances from C, and resolving each oblique force into a horizontal and perpendicular 
force (as in Prop. XVI.) it might be shown, by a similar manner of reasoning, that they would tend 
towards the earth with equal forces, and consequently, that an equilibrium would be produced. 

And if, instead of equal portions of the rod, portions of matter were placed at different distances, pjg, 6. 
which should be to each other inversely as those distances, as at F and B, the equilibrium would still 
be preserved ; for the forces with which such portions of matter, so situated, would endeavour to des¬ 
cend, would be equal, when the quantities of matter, multiplied into the velocities with which they are 
endeavouring to move, that is, into their distances (Prop. XI. Cor.) are equal: as will be more fully 
shown, in treating of the Mechanical Powers. 

Since, therefore, all the parts of any irregular body may be referred to some one of the above cases, 
it is manifest, that there is in every body a certain point, the parts on each side of w'hich balance each 
ether. 

PROP. XLVIII. If the centre of gravity in any body be supported, the whole body 
is supported ; if this centre be not supported, the body will fall. 

For when the centre of gravity is supported, the body rests on a prop on wfliich the parts on each 
side, acting with equal force against each other, will (Prop. XLV1I.) be in equilibrio, and neither side 
will move ; but when this centre is not supported, but the body has a prop under some other point, the 
parts of the body on one side of that other point will overbalance the parts on the other side, and the 
body will fall. 

Cor. Whenever a body moves by the power of gravitation, or falls, its centre of gravity descends; 
for if this centre do not descend, it must be supported ; and if the centre be supported, the whole body 
is sustained or kept from falling. 

Exp. 1. Let a board of a circular form be sustained perpendicularly on its centre of gravity, it will 
be at rest in any position. 

2. A beam, turning on an axis which passes through its centre of gravity, will rest in the same 
manner. 

3. A beam, whose axis passes through a point which is directly above the centre of gravity, will be 
at rest only when the beam is parallel to the plane of the horizon, because the centre of gravity will be 
*then tallen as low as possible. 

4. A cylinder, w hich has its centre of gravity near one of its sides, will roll up an inclined plane, if 
the side nearest the centre of gravity be placed towards the upper part of the plane; for this centre, 
Endeavouring to descend, wilLcarry the cylinder forward irf the ascending direction of the plane. 

5. Let a body, consisting of two equal and similar cones united at their bases, be placed upon the 


/ 


28 


OF MECHANICS. 


Book II, 


Plate 2. 
Fig. 7. 


Plate 2. 
Fig. 8. 


Ptate 2. 
-Fig. a. 


edges of two straight and smooth rods, which at one end meet in angle, and rest upon a horizontal plane, 
and at the other are raised a little above the plane, the body will roll towards the elevated end of the 
rules, and appear to ascend, while its centre of gravity descends ; as may be seen by applying a string 
horizontally above the path of the base of the cones. 

PROP. XLIX. If the line of direction come within the base on which any body is 
placed horizontally* the body will be sustained ; otherwise it will fall. 

In the body ABDE let C be the centre of gravity. The line of direction CO (that is, the line drawn 
from the centre of gravity towards the centre of the earth) being within the base DE, the body will bo 
supported, because the weight presses upon the base. Also since the body cannot fall towards K with¬ 
out turning round on the point E, the point C must in the motion ascend towards F, contrary to Prop. 
XLV1II. Cor. But in the position of the body a b d e, co the line of direction falling out of the base, c 
in its motion towards k descends, and the body will fall. 

Our own motions and actions are subject to this rule. When a man stands upright, his centre of 
gravity falls between his feet, and he is supported; but if he lean forward he throws the line of direc¬ 
tion without his base, and he would fall if he did not put forward one of his feet so as to cause it to fall 
within. Hence a porter, with a load on his back, leans forward that the load may not throw the line of 
direction out of his base behind; and by an artful adjusting of this point it is that such wonders are per* 
formed in horsemanship, and on the tight and slack rope, &c. 

Exp. 1. Any body of a cylindrical or other regular form, so placed upon its base, that its line of di¬ 
rection does not come within the base (which may be seen by a cord and freight suspended from the 
centre of gravity) will fall; otherwise it will not fall. 

2. Let two bodies be laid upon an inclined plane, the one a cube, the other a figure with many sides, 
and let the line of direction of the former fall within the base, and that of the latter without the base, 
the former will slide , the latter roll down the plane. 

Def. VII. The centre of motion is the point about which a body moves. 

PROP. L. A heavy body suspended on a centre of motion will be at rest* if the 
centre of gravity is directly under, or above, the centre of motion 5 otherwise it will 
move. 

If a heavy body E, hangs by a string on a centre of motion C, the action of gravitation at E is in 
the direction EL, contrary to the direction in which the string acts to prevent the body from falling. 
In this position, therefore, the opposite forces being equal and in contrary directions, destroy each other, 
and the body is at rest. But if the body is at p, one of the forces acts in the direction p C, and the 
other in the direction p L; that is, in directions oblique to each other; whence the body will move in 
the diagonal of the parallelogram formed by p C, p L. And in all cases, since (without the aid of me¬ 
chanical powers afterwards explained) the force which sustains any body must be equal to its weight, 
the centre of gravitation can only be at rest when these forces are in the same line of direction, that is, 
when the centre of gravity is directly under, or directly above the centre of motion. 

Exp. A circular board, sustained at a point above or below the centre of gravity, will only be at rest 
when the centre of gravity is at the lowest point, that is, in the line of direction ; or when the centre 
of gravity is in the same line above the centre of motion. 

Schol. If two or more bodies be united, they may be considered as one, and have a common centre 
of gravity. 

Exp. 1 . Let two unequal balls be fixed upon the ends of a wire, they will have a common centre of 
gravity. 

2 . A board, which of itself would fall from a table (its centre of gravity lying beyond the edge of 
the table) may be made, in the same position, to support a vessel of w T ater, hanging upon it near the 
table ; if a stick, fixed with one end at the bottom of the vessel, and the other in a hole in the horizon¬ 
tal board, be long enough to push the vessel a little out of the perpendicular; that is, to bring the 
centre of gravity of the whole under the table. 

Schol. The common centre of gravity of any number of bodies may be thus found. Let C be the 
common centre of gravity of two bodies, dividing their distances (see Prop. XLVII.) in such a manner, 
that AC is to CB, as B to A; whence A x AC = B x BC, and consequently if the point C is support¬ 
ed, the bodies A and B balance each other. Suppose a third body, equal to the sum of A and B, placed 
in their common centre of gravity C ; from the point C draw a right line to the centre of a third body 
D, which divide in O, so that OD may be to OC, as A -f B is to D ; then is O the common centre of 
the three bodies, A, B, D. In the same manner may be found the common centre of any number of 
"bodies. 




Chap. VI. 


OF THE MECHANICAL POWERS. 


29 


PROP. LI. If any number of bodies move uniformly in right lines, whether in the 
same or different directions, their common centre of gravity is either at rest, or moves 
uniformly in a right line. 

If two bodies, A and B, move towards each other in the same right line, having their common cen¬ 
tre of gravity C, and their momenta equal, the velocity of A will be to that of B, as the body B to the 
body A, that is (as was shown, Prop. XLVII.) as AC to BC. Whence (Prop. VI.) whilst A passes through 
AC, B will pass through BC, and the bodies will meet in C, which is their centre of gravity during 
their motion, and at the time of concourse ; therefore the point C remains at rest. 

In the same manner, it may be shown, that if the bodies recede from each other with uniform mo¬ 
tions, the centres of gravity will be at rest. 

Next, suppose that two bodies, A and B, move in different directions AC, BD, describing equal spaces 
AC, CE, and BD, DF, in equal times; their common centre of gravity L will move uniformly in a 
right line. Produce CA, DB, till they meet in G ; make AG to GH, as AC is to BD; draw the line 
AH ; and through C and E, draw Cl, EK, parallel to AH. AC is to HI (El. VI. 2 .) as AG to GH ; that 
is, as AC to BD. Therefore (El. V. 9.) HI is equal to BD, and adding IB to each, HB is equal to ID. 
In like manner, CE is to IK, as AG to GH; that is, as AC to BD, or CE to DF; therefore (El. V. 9.) 
IK is equal to DF, and adding KD to each, ID is equal to KF ; but ID was proved to be equal to HB; 
therefore, KF is equal to HB. From L, the common centre of gravity of the bodies A and B, draw LM 
parallel to BD; draw GM, and produce it till it cut Cl, EK, in the points N and O ; and through these 
points draw NP, OQ, parallel to BD. ALis to LB (El. VI. 2.) as AM to MH ; and CP to PD, (as CN to NI, 
that is) as AM to MH; therefore (El. V. 11.) CP is to PD, as AL to LB ; that is, (because L is the com¬ 
mon centre) as B to A. Consequently, P will be the common centre of the bodies when they are found 
in C and D ; and, in like manner, it may be shown, that Q will be their common centre, when they are in 
E and F. But, since ML is to HB, as AM to AH ; that is, as CN to Cl, that is, as NP to ID, and that 
HB has been proved to be equal to ID, ML is equal to NP ; and, in like manner, NP equal to OQ; 
whence (ML, NP, OQ, being parallel to one another) the line LPQ is equal to the line MNO, and the 
points, P, Q, (any points of the line in which the common centre of gravity is found as the bodies are 
moving from A to E, and from B to F) will be in a right line. Moreover, since (El. VI. 2.) AC is to 
CE, as MN to NO, or LP to PQ, and that AC is equal to CE, LP will be equal to PQ. Therefore the 
common centre of gravity of the bodies A and B, is always in the same right line, and moves uniformly, 
or passes over equal spaces in equal times. 

In like manner, the common centre of these two bodies and any third body, or of the three bodies 
and a fourth, &c. being found, it may be proved that it moves uniformly in a right line. 

Cor. 1. Hence it is manifest, that any forces acting upon a system of bodies, must affect the motion 
of the common centre of gravity of that system, in the same manner as if the same force were similarly 
applied to a body equal to the sum of all the bodies, placed in the common centre of gravity. And the 
mutual actions of the parts of a system upon each other, producing (by Prop. III.) equal momenta in con¬ 
trary directions, cannot change the state of motion or rest of their common centre of gravity. Conse¬ 
quently the law of a system of bodies, as to motion or rest, is the same as that of one body, and is rightly 
estimated from the motion of its centre of gravity. 

Cor. 2. Hence the centre of gravity of a system of bodies will not be disturbed by their mutual at¬ 
tractions, as the motions thus communicated are always equal and opposite. Hence the centre of grav¬ 
ity of our system of planets is either at rest, or moves uniformly in a straight line. The latter is sup¬ 
posed by Dr. Herschel to be the case. 


CHAPTER VI. 

Of Motion as directed by certain instruments , called Mechanical Powers. 

Def. VIII. That body, which communicates motion to another, is called the Power. 
Def. IX. That body, which receives motion from another, is called the Weight. 

Def. X. The Lever is a bar, moveable about a fixed point, called its fulcrum , or 
prop. It is in theory considered as an inflexible line without weight. It is of three 
kinds ; the first, when the prop is between the weight and the power; the second, when 


Plate 2. 
Fig. 5. 


Plate 2. 
Fig. 9. 



30 


OF MECHANICS. 


Book. II. 


Plate 2. 
Fig. 10. 


Plate 3. 
Fig. 1. 


Plate 2. 
Fig. 11. 


the weight is between the prop and the power ; the third, when the power is between 
the prop and the weight. 

Exp. Let the three kinds of the lever be shown, as in Plate 2 , Fig. 12 , 13, 14. 

PROP. LII. A power and weight acting upon the arms of a lever will balance each 
other, when the weight is to the power, as the perpendicular distance of the line in 
which the power acts, from the fulcrum, is to the perpendicular distance of the line in 
which the weight acts, from the fulcrum. 

Case. 1. When the power acts perpendicularly; Let AB be the lever, C the prop, P the power, W 
the weight. The force with which any body moves being as its momentum, (Prop. XIII.) and its mo¬ 
mentum as the quantity of matter multiplied into the velocity, (by Prop. XI. Cor.) the force with which 
the weight W would move in the first instant of its motion, if no other body counteracted it, would be 
as its quantity of matter multiplied into its velocity. But because the weight W is suspended from the 
lever AB at the point B, it would move with the same velocity as this point; which (as was shown in 
Pro}). XLVI1.) is as the distance of the point B from the prop C, or D. The force therefore, with 
which the weight W would move without any counteracting force, is as its quantity of matter multiplied 
into the distance of the point of suspension B from the prop C, or D. But the weight will be prevented 
from descending, if a force equal to that with which it would descend without obstruction, acts upon it 
in the contrary direction; that is, if a force be applied to raise the point B of the lever AB, equal to 
to that with which the weight W would draw it downwards. Let the power P be suspended from the other 
extremity of the lever at the point A ; and let the quantity of matter in the power P multiplied into the 
distance of A, its point of suspension from C, or D, be equal to the quantify of matter in the weight W 
multiplied into the distance of B from C, or D ; it appears from what has been said concerning the 
weight, that the force with which the power P, without obstruction, would descend and draw down the 
point A, is equal to the force with which the weight W would descend and draw down the point B. 
But, as much force as the power P exerts to draw down the point A, it exerts to raise the point B. 
Therefore equal and opposite forces are exerted to raise and depress the point B ; and consequently it 
will continue at rest, and the weight and power will balance each other. 

Case 2. When the power acts obliquely; 

Let the weight A hang freely fi’om one end of a balance, so as to have its line of direction DA per¬ 
pendicular to the arm of the balance ; and let another weight as B, be hung at the other end E, in such 
manner that its line of direction EC, by passing over a pulley at C, may be oblique to the arm of the 
balance. If the whole force of gravity in the weight B acting in the direction EC, be denoted by the 
line EC, it may be resolved into two forces denoted by EF and FC, acting in the directions of these 
lines; of which two forces, the latter only, which acts in the direction FC perpendicular to the arm of 
the balance, resists the force of gravity in the weight A, the other force FE acting in the direction of 
the line of the lever. Since, therefore, that part of the weight B which acts in opposition to the 
weight A, is to the who le weight B, as FC to EC ; it is manifest, that in order to make the weight B 
balance the weight A, it must exceed the weight A, in the same ratio that the line EC exceeds the line 
FC. If from G, the centre of motion, be let fall GH perpendicular to EC produced, that line will he the 
perpendicular distance of the direction EC from G ; and EG, equal to DG, the perpendicular distance 
qf the direction DA ; but the triangles EFC and EHG are similar, consequently (El. VI. 4.) as EC is to 
CF, so is EG to HG ; but the weight B is to the weight A, as EC to FC ; B is therefore to A, as EG, or 
DG, to IIG. 

Or more generally, Let C be the centre of motion in the lever KL ; let A and B be any two powers 
applied to it at K and L, acting in the directions KA and LB. From the centre of motion C, let CM 
and CN be perpendicular to those directions in M and N; suppose CM to be less than CN, and from 
the centre C, at the distance CN, describe the circle NIID, meeting KA in D. Let the power A be 
represented by DA, and let it be resolved into the power DG, acting in the direction CD, and the 
powder DF perpendicular to CD, by completing the parallelogram AFDG. 'The power DG, acting in 
the direction CD from the centre of the circle, or wheel, DIIN towards its circumference, has no 
effect in turning it round the centre, from D towards H, and tends only to carry it off from that centre. 
It is the part DF only that endeavours to move the wheel from D towards H and N, and is wholly 
employed in this effort. The power B may be conceived to be applied at N as well as at L, and to be 
wholly employed in endeavouring to turn the wheel the contrary way, from N towards H and D. If, 
therefore, the power B be equal to that part of A which is represented by DF, these efforts, being equal 
and opposite, must destroy each other’s effect; that is, when the power B is to the power A, as DF to 
DA, or (because of the similarity of the triangles AFD, DMC) as CM to CD, or as CM to CN, then the 


i 


Chap. VI. 


OF THE MECHANICAL POWERS. 


powers must lie in equilibrio; and those powers will sustain each other, which are inversely as the 
distances of their directions from the centre of motion. 

Sciiol. It is evident that in the first kind of lever, either the weight may exceed the power or the 
power may exceed the weight; but in the second kind, the weight must exceed the power, and in the 
third, the power must exceed the weight. The second is adapted to produce a slow motion by a swift 
one ; and the third serves to produce a swift motion of the weight, by a slow motion of the power. See 
Fig. 12, 13, and 14. 

To the first kind of lever may be reduced several sorts of instruments; such as the steelyard, whose 
arms are unequal; the false balance, whose arms are imperceptibly unequal; the common balance, 
whose accuracy depends on its possessing the following properties; (1.) The arms must be equal in 
length and weight. (2.) The centre of motion must be a little above, and directly over the centre of 
gravity. (3.) The points from which the scales are suspended should be in a right line, passing through 
the centre of gravity of the beam. And (4.) the friction of the beam on the centre of motion should be 
as little as possible. Scissars, pincers, snuffers, &c. arc formed of two levers, the fulcrum of which is 
the pin which rivets them. 

To the second kind of lever may be reduced oars and rudders of ships ; cutting knives fixed at one 
end ; doors moving on hinges, &c. 

To the third kind, we may refer the action of the muscles of animals, ladders fixed at one end v and 
raised against a wall by a man’s arms, &,c. 

D ep. XI. The wheel and axis is a wheel turning round together with its axis ; the 
power is applied to the circumference of the wheel, and the weight to that of the axis, 
by means of cords. 

PROP. LIII. An equilibrium is produced in the wheel and axis, when the weight 
is to the power, as the diameter of the wheel to the diameter of the axis. 

Let AB be the diameter of the wheel, DE that of the axis, W the weight, and P the power, sus- Plate 
pended from the points D and B. When the wheel has performed one revolution, the power P has **£•• 
drawn oft’as much chord from the wheel as is equal to its circumference, and has therefore moved 
through a space equal to that circumference. In the same time the weight W is raised through a 
space equal to the circumference of the axis, upon which the cord, by which the weight is suspended, 
is once turned round. Therefore the velocity of the power exceeds the velocity of the weight, as much 
as the circumference, that is, the diameter of the wheel exceeds that of the axis. If then the weight 
exceeds the power as much as the velocity of the power exceeds that of the weight, that is, as much 
as the diameter, or semidiameter of the wheel, AB, or CB, exceeds the diameter, or semidiameter 
of the axis, DE, orCE, the momenta will be equal, and the power and weight will balance each other. 

Or thus ; The axis and wheel is a lever of the first kind; in which the centre of motion is in C, the 
centre of the axis ; the weight W, sustained by the rope DW, is applied at the distance DC, the radius 
of the axis ; and the power P, acting in the direction PB perpendicular to CP>, the radius of the wheel, 
is applied at the distance of that radius; therefore, Prop. LII. there is an equilibrium, when the power 
is to the weight, as the radius of the roller to the radius of the wheel. 

Cor. 1 . Hence it is evident, that by increasing the diameter of the wheel, or diminishing that of the 
axis, a less power may sustain a given weight. 

Cor. 2. The thickness of the rope to which the weight is suspended, ought not to be neg¬ 
lected. 

Scjtol. To the wheel and axle we may refer the capstan, mills, cranes, &c. A drawing and descrip¬ 
tion of a safe and truly excellent crane, invented by Mr. James White, may be seen in the 10th volume 
of the Transactions of the Society for encouraging Arts and Sciences, in London. 

Def. XII. The pulley is a small wheel, moveable about its axis, by means of a 
cord, which passes over it. 

PROP. LIV. When the axis of the pulley is fixed, the pulley only changes the 
direction of the power; if moveable pulleys are used, an equilibrium is produced, 
where the power is to the weight as one to the number of ropes applied to them. If 
each moveable pulley has its own rope, each pulley will double the power. 

If the pulley ED be fixed upon the beam A, (be power and weight, in equilibrio, will be equal. Plate 
But, if one end of the rope be fixed in B, and the other supported by the power P, it is evident, that in f jg- : 
order to raise the weight W one foot, the power must rise two ; for both the ropes .BC and CP will be 11 °' ‘ 


32 


OF MECHANICS. 


Book II. 


Fig. 5. 


Fig. 6. 


Plate. 12 
Fig. 1. 


Plate 3. 
Fig. 7. 


PLate 3. 
Fig. 8. 


shortened a foot each ; wjience the space run over by the power will be double of that of the weight; 
if therefore the power be to the weight as 1 to 2, their momenta will be equal. For the same reason 
if there be four ropes passing from the upper to the lower pulleys, the velocity of the power will be 
quadruple of that of the weight, or as 4 to 1, &c. In all cases, therefore, when the pow r er is to the 
weight, as 1 to the number of ropes passing from the upper to the lon r er pulleys, there will be an 
equilibrium. 

Or thus; Every moveable pulley hangs by two ropes equally stretched, which must bear equal parts 
of the weight; and therefore when one and the same rope goes round several fixed and moveable 
pulleys, since all its parts on each side of the pulleys are equally stretched, the w hole weight must bo 
divided equally amongst all the ropes by which the moveable pulleys hang. Consequently, if the 
power which acts on one rope be equal to the weight divided by that number of ropes, the powder must 
sustain the weight. 

If each moveable pulley has its own cord, the first, as appears from what has been said, doubles the 
velocity of the power; anil therefore if the power be half of the weight, the momenta will be equal, 
and the balance will be produced. In like manner, the second puiley causes the weight to move with 
half the velocity with which it would move, if suspended from the first moveable pulley, that is, makes 
the velocity of the power quadruple of that of the weight; and so of the rest. 

If in the solid block B, grooves be cut, whose radii are 1, 3, 5, 7, &c. and in the block A other 
grooves be cut, whose radii are 2, 4, 6, 8, &c. and a string be fastened to A and passed round these 
grooves, the grooves will answer the purpose of so many distinct pulleys, and every point in each, 
moving with the velocity of the string in contact with it, the whole friction will be removed to the two 
centres of motion in the blocks A and B, which is a great advantage over the common pulleys. This 
pulley was invented by Mr. James White. 

PROP. LV. In the inclined plane the power and weight balance each other, when 
the power is to the weight, as the sine of the inclination of the plane is to the sine of 
the angle, which the line of the direction of the power makes with the perpendicular 
to the plane. 

Let a weight be supported on the inclined plane CA by a power acting in any given direction PD. 
Let the whole force, whereby the weight would descend perpendicularly, be represented by BP ; and 
resolving PB into tw r o forces, one of which, BD, is perpendicular to the plane CA, and the other, PD, is 
in the direction of the power; the force BD is destroyed by the reaction of the plane, and the force 
PD will be sustained by an equal power, acting in the direction PD. Therefore, when there is an 
equilibrium, the power is to the weight, as PD to PB ; that is, as the sine of the angle PBD, or (El. 
VI. 8.) its equal CAB, to the sine of the angle PDB. 

When PD is in the direction of the plane, this ratio becomes that of CD to CB, or of the height of 
the plane CB, to CA its length. 

When the direction of the power PD is parallel to the base of the plane, the ratio of the power to 
the weight becomes that of ED to EB; or (El. VI. 8. Cor.) of CB, the height of the plane, to BA, 
the base. 

When the direction of the power coincides with the perpendicular BD, the ratio of the power to 
the weight becomes that of the sine of a finite angle, to the sine of an angle indefinitely diminished. 
From which it appears, that no finite power is sufficient to support a weight upon an inclined plane, if 
that power acts in a direction perpendicular to the plane. 

Def. XIII. The screw is a cylinder, which has either a prominent part, or a hollow 
line, passing round it in a spiral form, so inserted in one of the opposite kind, that it 
may be raised or depressed at pleasure, with the weight upon its upper, or suspended 
beneath its lower, surface. 

PROP. LVI. In the screw the equilibrium will be produced, when the power is to 
the weight, ds the distance between two contiguous threads, in a direction parallel to 
the axis of the screw, to the circumference of the circle described by the power in one 
revolution. 

While the screw is made to perform one revolution, the weight W may be considered as raised up 
an inclined plane c </, whose height cp is the interval between two contiguous spirals, whose base p q is 
the periphery of the cylinder, and wffiose length c q is the spiral line, by a power acting parallel to the 


Chap. VI. 


OF THE MECHANICAL POWERS. 


33 


base of the plane; for such an inclined plane, involved about a cylinder, will form the spiral line of the 
screw. A power at£>, acting parallel to the base, is in equilibrio with the weight W to be raised, when 
the power is to the weight, as the height of the inclined plane, to the base; or in this case as p c, the 
interval between the spirals, to the circumference described by p ; but a power applied at P, which is 
to that applied at y?, as the circumference described by p, to the circumference described by P, has the 
same effect; therefore there is an equilibrium, when the power applied at P is to the weight to be raised, 
as p c, the interval between two contiguous spirals, to the circumference described by the power P. 

Dei\ XIV. The wedge is composed of two inclined planes, whose bases are 
joined. 

PROP. LVII. When the resisting forces, and the power which acts on the wedge, 
are in equilibrio, the weight will be to the power, as the height of the wedge, to a 
line, drawn from the middle of the base to one side, and parallel to the direction in 
which the resisting force acts on that side. 

Let the equilateral triangle ABC represent a wedge, whose base, or back, is AC, whose sides are 
the lines AB and CB, and whose height is the line BP, which bisects the vertical angle ABC, and also 
the base perpendicularly in P. Let E and F represent two bodies, or two resisting forces acting on the 
aides of the wedge perpendicularly, and whose lines of direction EP and FP meet at the middle point of 
the base, on which the power P acts perpendicularly, then will EP and FP (El. I. 5, and 26) be equal. 
Let the parallelogram ENFP be completed ; its diagonals PN and EF will bisect each other perpendic¬ 
ularly in H. Now'when these forces (which act perpendicularly on the sides and base of the wedge) 
are in equilibrio, they will be to each other (Prop. XIV.) as the sides and diagonal of this parallelo¬ 
gram; that is, the sum of the resisting forces will be to the pow er P, as the sides EP and FP to the 
diagonal PN, or as one side EP to half the diagonal PH ; that is, (from the similarity of the right- 
angled triangles BEP, EHP) as BP, the height of the ^edge, to EP, the line which is drawn from 
the middle of the base to the side AB, and is the direction in which the resisting force acts on 
that side. 

From the demonstration of this case, in which the resisting forces act perpendicularly on the sides 
of the wedge, it appears that the resistance is to the power which sustains it, as one side of the wedge 
AB is to the half of its breadth AP; because AB is to AP, (El. VI. 8.) as BP is to EP. 

It appears also from hence, that if PN be made to denote the force with which the power P acts on 
the wedge, the lines PE and PF, which are perpendicular to the sides, will denote the force with which 
the power P protrudes the resisting bodies in directions perpendicular to the sides of the wedge. 

Let us now suppose, in the second case, that the resisting bodies E and F act upon the wedge in di¬ 
rections parallel to the lines DP and OP, which are equally inclined to ils sides, and meet in the point 
P. Draw the lines EG and FK perpendicular to DP and OP ; then making PN denote the force with 
which the powder P acts on the wedge, PE and PF will denote the forces with which it protrudes the 
resisting bodies in directions perpendicular to the sides of the wedge, as was observed before; now 
each of these forces may be resolved into two, denoted respectively by the lines PG and GE, PK and 
KF, of which GE and KF wili be lost, as they act in directions perpendicular to those of the resisting 
bodies ; and PG and PK will denote the forces by which the power P opposes the resisting bodies, by 
protruding them in directions contrary to those in which they act on the wedge; therefore, when the 
resisting forces are in equilibrio with the power P, the former must be to the latter, as the sum of the 
lines PG and PK, is t to PN, or as PG is to PH. But (El. VI. 4.) PG is to PE, as PE to PD ; and PH is 
to PE, as PE to PB; whence (El. VI. 16.) both the rectangle PG x PD and the rectangle PH x PB, 
are equal to the square of PE; these rectangles are therefore equal to one another; whence their 
sides (El. VI. 14.) are reciprocally proportional; that is, PG is to PH, as PB to PD. Whence it follows 
from what was shown above, that, in equilibrio, the resisting forces are to the power, as PB to PD; 
that is, as the height of the wedge to the line drawn from the middle of the base to one side of the 
wedge, and parallel to the direction in which the resisting force acts on that side. 

From what has been demonstrated, we may deduce the proportion of the power to tire resistance it 
is able to sustain in all the cases in which the wedge is applied. First, when in cleaving timber the 
wedge fills the cleft, then the resistance of the timber acts perpendicularly on the sides of the wedge ; 
therefore in this case, when the power which drives the wedge, is to the cohesive force of the timber, 
as half the base, to one side of the wedge, the power and resistance will be in equilibrio. 

Secondly; When the wrndge does not exactly fill the cleft, which generally happens because the 
wood splits to some distance before the wedge; let ELF represent a cleft into which the w'edge ABC 
is partly driven : as the resisting force of the timber must act on the wedge in directions perpendicular 

5 


Plate 3. 
Pig. 9. 


34 


OF MECHANICS. 


Book II 


Plate 2. 
Fig. 15. 

Fig. 16. 


to the sides of the cleft, draw the line PD in a direction perpendicular to EL, the side of the cleft, and 
meeting the side of the wedge in D ; then the power driving the wedge and the resistance of the tim¬ 
ber, when they balance, will be to each other as the line PD to PB, the height of the wedge.* 

Thirdly; When a wedge is employed to separate two bodies that lie together on a horizontal plane, 
for instance, two blocks of stone; as these bodies must recede from each other in horizontal directions, 
their resistance must act on the wedge in lines parallel to its base CA; therefore the power which 
drives the wedge will balance the resistance when they are to each other as PA, half the breadth of 
the wedge, to PB its height. 

Schol. 1 . Since in all the mechanical powers, an equilibrium is produced, when the power is to the 
weight as the velocity of the weight is to the velocity of the power, in all oompound machines there 
will be an equilibrium, when the sum of the powers is to the weight, as the velocity of the weight its 
to the sum of the velocities of the powers. 

Schol. 2. In the theory of mechanical powers, we suppose all planes and bodies perfectly smooth ; 
levers to have no weight; cords to be perfectly pliable, and the parts of machines to have no friction. 
(See Schol. 3.) Allowances, however, must be made for the difference between theory and practice. 
Mr. Ferguson observes, that there are but few compound machines, but what, on account of friction, 
will require a third part more to work them, when loaded, than w hat is sufficient to constitute an equi¬ 
librium between the weight and the power. 

Exp. 1. Let A, B, C, be a compound lever, consisting of three levers, in the first of which, A, the 
velocity of the w r eight is to that of the power, as 1 to 5 ; in the-second, B, as 1 to 4; in the third, C, 
as 1 to 6 . The velocity of the weight will be to that of the power, as 1 to 5x4x6== 120 ; and if 
the power be to the w’eight, as 1 to 120 , they will balance each other. 

2. Let GC and LF be the levers fixed to the supporters RA, SE, and let their shorter arms be kept in 
equilibrio with the longer respectively by the weights fixed at G and L. Let NH be a bar screwed to 
the fixed parts to keep them steady. If the power C be ten times farther from A the prop, than the 
weight P, they will be in equilibrio when the power C is to the weight P, as 1 to 10. In like manner, 
the distance ME being ten times DE, if the power M be -i- of the weight C suspended from D, they 
will be in equilibrio; whence M, 1, will balance P, 100 . 

3. Exhibit models or draughts of different compound machines, as mills, cranes, the pile-driver, &c. 

Schol. 3. The inequality of the surface on which any body moves occasions an attrition, called fric¬ 
tion, which prevents the accurate agreement of many experiments in mechanics, with theory. On this 
subject the very accurate experiments of Mr. Vince should be consulted, the object of which was, to 
determine, ( 1 .) Whether friction be an uniformly retarding force. ( 2 .) The quantity of friction. (3.) 
Whether friction varies in proportion to the pressure or weight. And (4.) whether the friction be the 
same, on whichever of its surfaces a body moves. After a great variety of experiments made with the 
utmost care and attention, Mr. Vince deduces the following conclusions, which may be considered as 
established facts. 

1. That friction is an uniformly retarding force in hard bodies, not subject to alteration by the ve¬ 
locity ; except when the body is covered with woollen cloth, &,c. and in that case the friction increases 
a little with the velocity. 

2 . Friction increases in a less ratio than the weight of the body, being different in different bodies. 
It is not yet sufficiently known for any one body, what proportion the increase of friction bears to the 
increase of weight. 

3. The smallest surface has the least friction, the weight being the same. But the ratio of the 
friction to the surface is not accurately known. 

See a full account of these experiments, Vol. lxxv, Phil. Trans. 

Schol. 4. Wheel carriages are used, to avoid friction as much as possible. A wheel turns round 
upon its axis, because the several points of its circumference are retarded in succession by attri¬ 
tion, whilst the opposite points move freely. Large wheels meet with less resistance than smaller 
Irom external obstacles, and from the friction of the axle, and are more easily drawn, havin'*- 
their axles level with the horses. But in uneven roads, small wheels are used, that in ascents the ac¬ 
tion of the horse may be nearly parallel with the plane of ascent, and therefore may have the great¬ 
est effect; small wheels are also more conveniently turned. The greater part of the load should be laid 
on the hinder part of a wheel carriage. 

* In estimating the lateral cohesion of woody fibres when separated by a wedge, the pressure on only one side of the 
wedge should be reckoned, so that on this account the cohesion should be estimated at only half what it is in the text ; but 
on another account, not mentioned in the text, it should be reckoned much more, for the sides of the cleft are actually 
levers, in which the pressure of the wedge is the power, the point where the cohesion is just giving way is the place of 
weight, while the fulcrum is at some distance further from the wedge, more or less, according to the rigidity or flexibility 
of the timber; so^that, notwithstanding the cohesive force is erroneously doubled in the text, it^is probably much 
underrated. 


Chap. VII. 


OF PROJECTILES. 


35 


CHAPTER VII. 

Of Motion as 'produced by the united Forces of Projection and Gravitation. 

section i. 

Of Projectiles. 

PROP. LVIII. Bodies thrown horizontally or obliquely, have a curvilinear motion, 
and the path which they describe is a parabola; the air’s resistance not being consid¬ 
ered. 

If a body be thrown in the direction AF, and acted upon by the projectile force alone, it will con- Plate 3. 
tinue to move on uniformly in the right line AF, and would describe equal parts of the line AF in equal Fi S- 
times, as AC, CD, DE, &c. But if, in any indefinitely small portion of time, in which the body would by 
the projectile force move from A to C, it would, by the force of gravity, have fallen from A to G; by 
the composition of these forces (Prop. XVI.) it will at the end of that time, be found in H, the opposite 
angle of the parallelogram ACGH. In two such portions of time, whilst it would hare moved from A 
to D by the projectile force, it would (Prop. XXVI.) by gravitation fall through four times AG, that is, 

AM; and therefore, these forces being combined, it will be found at the end of that time in I, the oppo¬ 
site angle of the parallelogram DM. In like manner, at the end of the third portion of time, it would 
by the projectile force be carried through three equal divisions to E, and by the force of gravitation 
over nine times AG to N; and consequently, by both these forces acting jointly, it will be carried to K, 
the opposite angle of the parallelogram EN. Therefore the lines CH, Dl, EK ; that is, AG, AM, AN, 
which are to each other as the numbers 1, 4, 9, are as the squares of the lines AC, AD, AE; that is, 

GH, MI, NK, which are as 1,2, 3. And because the action of gravitation is continual, the body in 
passing from A to H, &c. is perpetually drawn out of the right line in which it would move if the force 
of gravitation were suspended, and therefore moves in a curve. And H, I, and K are any points in this 
curve in which lines let fall from points equally distant from A in the line AB meet the curve. There¬ 
fore the body moves in a parabola, the property of which is [Simpson’’s Conic Sectio7is , Book I. Prop. XII. 

Cor.) that the abscissce AG, AM, AN, are to each other as the squares ot the ordinates GH, MI, NK. 

Remark. Very dense bodies moving with small velocities describe the parabolic track so nearly, 
that any deviation is scarcely discoverable ; but with very considerable velocities the resistance of the 
air will cause the body projected to describe a path altogether different from a parabola, which will not 
appear surprising when it is known that the resistance of the air to a cannon ball of two pounds weight, 
with the velocity of 2000 feet per second, is more than equivalent to 60 times the weight of the ball. 

See Hutton’s Diet. Art. Resistance. 

PROP. LIX. The path which a body thrown perpendicularly upward describes 
in rising and falling is a parabola. 

A stone lying upon the surface of the earth, partaking of the motion of the earth (here supposed) 
round its axis, this motion which it has with the earth will not be destroyed by throwing it in a direc¬ 
tion perpendicular to the surface of the earth. After the projection, therefore, the stone will be moved 
by two forces, one horizontal, the other perpendicular, and will rise in a direction which may be shown, 
as in the last proposition, to be the parabolic curve ; in which it will continue till it reaches the high¬ 
est point, from whence it might be shown, as in the last proposition, that it will descend through the 
other side of the parabola. 

PROP. LX. The velocity with which a body ought to be projected to make it 
describe a given parabola, is such as it would acquire by falling through a space equal to 
the fourth part of the parameter belonging to that point of the parabola from which it 
is intended to be projected. 

The velocity of the projectile at the point A (by Prop LVIII.) is such as would carry it from A to E, p] ate 3 
in the same time in which it would descend by its gravity from A to N. And the velocity acquired in Fig. 10, 
falling from A to N (by Prop. XXVII.) is such as in the same time by an uniform motion would carry the U. 


36 


OF MECHANICS. 


Book II. 


Plate 3. 
Fig. 13. 


body through a space double of AN. Therefore the velocity which is acquired by the body in 
falling to N is to that with which the body is projected at A, and uniformly carried forward to 
E, as twice AN is to AE. But since, from the nature of the parabola ( Simpson's Conic Sections , Book I. 

A 17 * 

Prop. XIII.) is equal to the parameter of the point A, one fourth part of this parameter will be 


expressed by 


AN 

|AE* 

AN 


And because the velocities acquired by falling bodies are (by Prop. XXVI. Cor. 1.) 


as the square roots of the spaces they fall through, the velocity acquired by a body in falling through 

AAE* 

AN is to the velocity acquired in falling through V V - or one fourth part of the parameter of A, as the 


square root of AN to the squore root of 1 1 that is, as N / AN to ^ ^ i or AN to \ AE, or twice AN 

to AE. Therefore the velocity acquired by a body in falling from A to N has the same ratio to the ve¬ 
locity with which the body is projected or the line AE described, and to the velocity acquired by a body 
in falling through a fourth part of the parameter belonging to the point A; consequently (El. V. II.) 
these velocities are equal. 

Cor. Hence may be determined the direction in which a projectile from a given point, with a given 
velocit}', must be thrown to strike an object in a given situation. 

Let A be the place from which a body is to be thrown, and K the situation of the object. Raise AB 
perpendicular to the plane of the horizon, and equal to four times the height from which a body must 
fall to acquire the given velocity. Bisect AB in G ; through G draw HG perpendicular to AB ; at the 
point A raise AC perpendicular to AK, and meeting IlG in C ; on C as a centre with a radius CA de¬ 
scribe the circle ABD ; and through K draw the right line KE1 perpendicular to the plane of the hori¬ 
zon, and cutting the circle ABD in the points E and I. AE, or Al, will be the direction required. 

For, drawing BI, BE, since AK is a tangent to the circle, and BA, IK, are parallel to each other, the 
angle ABE (El. III. 32.) is equal to the angle EAK; and the alternate angles BAE, AEK, are equal; 
therefore the triangles ABE, AEK, are similar; and AB is to AE, as AE to EK. Therefore AB x EK 


= AE 2 ; and AB — 


AE* 
EK * 


In like manner, the triangles BAI, KAI, being 


similar, BA is equal to 


AD 


-. Since, then, AB is equal to four times the height from which a body must fall to acquire the 

llv 

AE 2 AI 2 

velocity with which it is to be thrown; (or its equal) is the same. Consequently (by this 


Prop.) the point K will be in the parabola which the body will describe, which is thrown with the 
given velocity in the direction AE, or AI, and the body will strike an object placed at K. 

Schol. If the velocity with which a projectile is thrown be required, it may be determined from 
experiments in the following manner. By the help of a pendulum or any other exact chronometer, let 
the time of the perpendicular flight be taken • then, since the times of the ascent and descent are equal, 
the time of the descent must be equal to one half of the time of the flight, consequently, that time 
will be known; and, since a heavy body descends from a state of rest at the rate of 16.1 feet in the first 
second of time, and that the spaces through which bodies descend are as the squares of the times ; if we 
say, as one second is to 16.1 feet, so is the square of the number of seconds which express the time of 
the descent of the projectile, to a fourth proportional, we shall have the number of feet through which 
the projectile fell, which being doubled, will give us the number of feet which the projectile would 
describe in the same time with that of the fall, supposing it moved with a uniform velocity, equal to 
that which it acquired by the end of the fall; which last found number of feet, being divided by the 
number of seconds which express the time of the projectile’s descent, will give a quotient, expressing 
the number of feet, through which the projectile would move in one second of time with a velocity 
equal to that which it acquired in its descent, which velocity is equal to the velocity with which the 
projectile w T as thrown up; consequently, this velocity is discovered. 4 


PROP. LXI. The squares of the velocities of a projectile in different points of its 
parabola, are as the parameters belonging to those points. 

For (by the last Prop.) the velocities in the several points of the parabola, are equal to the velocities 
acquired in falling through the fourth parts of the parameters of the points. Therefore the squares of 
these velocities being (by Prop. XXVI.) as the spaces described, the squares of the velocities in the 
several points of the parabola are as the fourth parts of the parameters of those points; but the whole 
parameters are as their fourth parts ; therefore the squares of the velocities at the several points of the 
parabola are as the parameters of those points. 










Chap. VII. 


OF PROJECTILES. 


37 


t 

Cor. Hence, setting aside any difference which may arise from the resistance of the air, a projectile 
will strike a mark as forcibly at the end as at the beginning of its course, if the two points be equally 
distant from the principal vertex ; for, the parameters belonging to these points being equal, the veloci¬ 
ties in these points must also be equal. 

PROP. LXII. When a body is thrown obliquely with a given velocity, if the space 
through which it must have fallen perpendicularly to acquire that velocity is made the 
diameter of the circle, the height to which the body will rise is equal to the versed 
sine of double the angle of elevation. 

Let a body be thrown in the direction BE, with the same velocity which any body would acquire pi a te 3. 
by falling perpendicularly through AB ; if AB be made the diameter of a circle, the greatest height to Fig. 12. 
which it will rise will be BD. 

Let IL be a right line drawn in the plane of the horizon, touching the circle in B, and making with 
the line BE, which is the direction in which the body is thrown, the angle IBE, or angle of elevation. 

Because IL touches the circle, and EB drawn in the circle meets it in the point of contact, (El. III. 32.) 
the angle EBI is equal to the angle EAB. And ECB is double of EAB, (El. III. 20.) therefore ECB is 
double of EBI, the angle of elevation. And BD is the versed sine of ECB ; that is, of double the angle 
of elevation. 

Let BE represent the velocity with which the body is thrown. Then since this velocity is, by sup¬ 
position, such as might be acquired by falling down AB, if the body was thrown perpendicularly upward 
with the same velocity BE, it would rise to the height BA. Let the oblique motion BE be resolved into 
two others, one in the direction BD perpendicular to the horizon, and the other in the direction DE 
parallel to it; then the ascending velocity will be to the horizontal velocity, as BD to DE, and to the 
whole velocity, as BD to BE. But the part of the velocity BD is the only part which is employed in 
raising the body, since the other part DE is parallel to the plane of the horizon. Now, the height of 
a body ascending perpendicularly with the whole velocity BE, will be to the height when it ascends 
with the partBD (compare Prop. XXVI. and Prop. XXVIII.) as the square of BE to the square of BD. 

But because (El. VI. 8.) the triangle EDB is similar to the triangle AEB, BD is to EB, as EB is to BA; 
antfBD, BE, BA, being continued proportionals, BD is to BA, as the square of BD is to the square of 
BE. And the perpendicular heights to which the velocities BE and BD will make the body ascend have 
been shown to be as the square of BE to the square of BD; the heights are therefore as BA to BD. 

Since therefore the first velocity BE would make the body ascend through BA, the other velocity BD, 
w r hich is the part of the whole velocity which acts to make the body thrown in the direction BE to as¬ 
cend, will carry it to the height BD, which is* the versed sine of double the angle of elevation. The 
same might be shown in any other direction of the body, as BF, or BG. 

Def. XV. The Random of a projectile is the horizontal distance to which a heavy 
body is thrown. 

PROP. LXIII. When a body is thrown obliquely with a given velocity, if the space 
through which it must have fallen perpendicularly to acquire that velocity is made the 
diameter of a circle, the random will be equal to four times the sine of double the angle 
of elevation. 

If EBI be the angle of elevation, and ECB double that angle, DE will be the sine of double the an- Plate 3, 
gle of elevation. Let a body be thrown from the point B in the direction BE, with the velocity which Fig. 
it would acquire in falling through AB; the random, or horizontal distance at which the body will fall, 
is equal to four times DE. 

For, since (as in the last Prop.) the velocity BE being resolved into BD, DE, the ascending velocity 
is BD, and the horizontal DE, if these two velocities were to continue uniform, the spaces described in 
equal times (Prop. V.) would be as the velocities, and in the same time in which the body by the as¬ 
cending velocity would rise through BD, by the horizontal velocity it would be carried forward through 
DE. Of these velocities, the horizontal one DE is uniform, because the force of gravity can nei¬ 
ther accelerate nor retard a motion in this direction; but the ascending velocity is uniformly retard¬ 
ed; and therefore the body (compare Prop. XXIII. and XXVIII ) will be twice as long in ascending 
to its greatest height BD, as it would have been if the first ascending velocity had continued uniform; 
but on this supposition, the body would have been carried through BD and DE in the same time ; there¬ 
fore in double the time, that is, in the time of ascent through BD with an uniformly retarded velocity, 
it would be carried forward through twice DE; consequently, la the times of descent and ascent togeth¬ 
er it would move forward through four limes DE. Therefore a body thrown from B in the direction. 


38 OF MECHANICS. Book II. 

BE with such a velocity as might be acquired by falling down AB, the diameter of a circle, will fall at 
the distance of four times the sine of double the angle of elevation. 

PROP. LXIV. The random of a projectile will be the greatest possible, with a giv¬ 
en velocity, when the angle of elevation is an angle of forty-five degrees. 

The velocity being given, the height from whence the body must have fallen to acquire that velocity, 
or (Prop. XXXV.) the diameter of the circle AB, is a given quantity. And in a given circle the great¬ 
est sine is the radius or sine of a right angle; therefore four times the radius is greater than four times 
any other sine ; and consequently, the random which is equal to four times the radius (which by Prop. 
LX1I. will be the case when the double angle of elevation is a right one, or the angle of elevation forty- 
live degrees) will be the greatest possible random. 

Exp. This proposition, and the two following, may be illustrated by water spouting from a pipe. 

PROP. LXV. The random of a projectile, whose velocity is given, will be the 
same at two different elevations, if the one be as much above forty-five degrees as the 
other is below it. 

Plate 3. If EBI be an angle of 30 degrees, and GBI an angle of GO degrees, because EBI falls short of half a 

Fig. 12. right angle as much as GBI exceeds it, the double of EBI will fall short of a right angle as much as the 

double of GBI will exceed it; therefore, from the definition of a sine, these doubles will have the same 
sine. Consequently, four times their sines, that is, (by Prop. LXI1I.) their randoms will be equal. 

PROP. LXVI, The greatest random of a projectile, whose velocity is given, is 
double the height to which it would rise if it were thrown perpendicularly with the 
same velocity. 

Plate 3. If ^ body be projected in the direction BF, at an angle of forty-five degrees, and its velocity be equal 

Fig. 12. to that which a body would acquire in falling down AB, (by Prop. LXIV.) the random will be the greatest 
possible, and will be equal to four times CF, or twice BA. But the body cast perpendicularly" upwards 
with the same velocity would (by Prop. XXVIII.) rise to the height BA. Therefore the greatest random, 
with a given velocity, is double the height to which the body, thrown perpendicularly with the same 
velocity, would rise. 

PROP. LXVII. The randoms of projectiles, whose elevations are given, are as the 
squares of their velocities. 

Plate 3. If a body be thrown in any direction BE, its random (Prop. LXIII.) will be equal to four times DE, 

*'*g 12* or four times the sine of double the angle of elevation, in a circle whose diameter AB is the height 

from which the body must fall to acquire the velocity with w’hich it is projected. And because, in the 
triangle EDC, the angle at D being a right angle is always invariable, and that the angle ECD, which is 
double of EAD, that is, (El. III. 32.) of the given angle of elevation EBI, is given, the triangle ECD in 
every variation of AB, is always equiangular and similar to itself, and ED is always as EC ; but EC being 
a radius, is as AB; therefore ED, the sine of twice the given angle of elevation, is as AB, the diameter. 
Consequently four times the sine ED, that is, the random, is as AB. But the height AB, from which a 
body must fall to acquire any velocity, is (by Prop. XXVI.) as the square of that velocity. Therefore 
the random is as the square of the velocity. 

SECTION II. 

Of Central Forces. 

PROP. LXVIII. A body which is constantly drawn or impelled toward any point, 
may be made to describe, round that point as a centre, a curve returning into itself. 

Plate 3. Let ^ -^ e centre °f the earth, and GDEI its surface. Let a body be projected in any direction 

Fig. 14. GH, which does not pass within the surface of the earth. The projectile force, together with the 
force of gravity, will make it describe a curve, w'hich, as the projectile force is increased, will recede 
farther from the perpendicular GE, as GB, GC, GD. It is manifest that the projectile force may be 
increased, till the body shall pass beyond the surface CDKF. and move in the path GML, GNV, or 
some larger curve 


Ciiap. VII. 


OF CENTRAL FORCES. 


First; Suppose the projectile force to be such, that the body will be carried in the semicircle GN, 
it will continue in the curve of that circle till it returns to G. For, when a body moves in the circum¬ 
ference of a circle (as in fig. 15.) the projectile force, acting in a line which is a tangent to the circle, 
as GB, acts (El. III. 18.) in a direction which is perpendicular to the direction BA, in which it is ira- 
pelled towards the centre. And since if the force which impels the body towards the centre ceased to 
act in any point, as C, the body would move forward in the right line CF, the projectile force in every 
point of the circumference, acts in a direction perpendicular to the force of gravitation; consequently, 
these two forces remaining the same, and acting always in the same direction with respect to each 
other, the velocity of the body must remain the same ; whence, at the point M, it will have the same Fig. 14. 
power to recede from the centre as at G; and, retaining this power through every remaining part of its 
course, it will proceed in the circumference, till it arrive at G, and will continue to revolve in the circle. 

Next; Let the body be projected from G with a force less than that which is required to carry it 
round in the circumference of the circle GNV; and let the curve in which it moves be an ellipse, hav¬ 
ing the earth in its remoter focus. Because the force of projection, as the body proceeds in the first 
half of its orbit, acts in the direction of a tangent to the curve, whilst the force of gravitation acts in 
the direction of a right line from the body to the centre of the earth, the directions of these two forces 
make an acute angle with one another, and consequently, through this part of the course of the body, 
the force of gravitation conspiring -with the force of projection, the velocity of the body must be increased, 
and at the same time it must be continually drawn downward toward the earth. At the point in which 
the forces act in directions perpendicular to each other, the force of gravitation does not conspire with 
that of projection to bring the body toward the earth; and afterward in the latter half of its course, the 
directions of the forces making an obtuse angle with each other, the force of gravitation is opposed by 
that of projection in the same degree in which the former was before aided by the latter; and there¬ 
fore the body in passing toward G will fly off from the earth or rise, as much as it before approached to 
the earth or descended , and thus will return to the point G with the same velocity with which it set out 
at first, having lost as much velocity by receding from the earth in the latter part of its course, as it 
had gained by falling toward the earth in the former part. 

Lastly; Let the body be projected from G with a force which is greater than sufficient to carry it 
round in the circle GNV ; and let it perform its revolutions in an elliptic curve, whose greater axis is 
greater than the diameter of the circle GNV, setting out from G, and having the earth in the nearer 
focus; the effect will be the same as in the last case, except that the projectile force will oppose the 
force of gravitation in the first half of the revolution, and conspire it with the latter. 

Exp. Let a ball revolve round the central point of a whirling table. Concerning the construction 
and use of this machine, see Ferguson’s Lectures, Lect. II. 

PROP. LXIX. A body revolving in an orbit, endeavors in every point of its course 
to fly off from the centre in a right line, which is a tangent to the orbit. 

Let BCDL be a circle in which a body is revolving; when it is arrived at the point B, let the p] ate 3 
force which impels it toward the centre be withdrawn, and the body (by Prop. I.) would fly off from the Fig. 15 
point B in the direction BG ; in like manner at C, it would fly off in the right line CF; at D, in DH; 
and at L, in LK. The same is manifestly true in an elliptical orbit. Now the same force with which it 
would fly off, if no other cause prevented it, must make it endeavour to fly off in the same manner in 
every point of the orbit. 

Exp. Whilst a ball is revolving on a whirling table, if the cord which retains it be suddenly cut, the 
ball will fly off in a right line, which will be a tangent to the orbit in which it moved. 

Cor. A body revolving about a centre endeavours to recede from that centre ; for every point of 
the tangent in which it endeavours to move out of the circle, is farther from the centre, than the point 
in which the tangent meets the curve. 

Def. XVI. The force which impels a body toward the centre, when it revolves in 
an orbit, is called the centripetal force ; that by which it endeavours to recede from the 
centre, is called the centrifugal force; and these two forces are called jointly the cen¬ 
tral forces. 

Schol. The projectile and centrifugal forces differ from each other, as the whole from the part. 

The projectile force is that with which a body would move forward in a tangent to its orbit, if there 
were no centripetal force to prevent it; the centrifugal force is that part of the projectile force which 
carries the body off from the centre while it is describing the tangent. Thus if the body revolved in P ] ate3 
the orbit BD, the projectile force is that which would make it describe the tangent BA, if the centripetal Fig. ig. 
force were to cease acting. But in the mean time, the whole force BA does not carry the body off 


40 


OF MECHANICS. 


Book II. 


Plate 3. 
Fig. 16. 


Plate 3. 
Fig. 8. 


Plate 3. 
Fig. 18. 


from the centre C ; when it is arrived at A, it is farther from the centre than it was at B, only by the 
length AN, and it is that part of the projectile force which, when the whole is resolved into two forces, 
may be considered as acting in this line AN, which carries the body off from the centre, and is called 
the centrifugal force. 

PROP. LXX. When bodies revolve in a circular orbit about a centre, the cen¬ 
tripetal and centrifugal forces are equal. 

If a body revolve in the circle BD, in the time in which it describes the arc BN, it will have been 
impelled tow'ard the centre through the space AN; for, by the projectile force alone it would have 
been carried from B to A. The line ^N is then the space described by means of the centripetal force, 
and this force is proportional to AN. But if, when the body was at B, no centripetal force had acted 
upon it, instead of describing the arc BN, it would have moved along the tangent BA, and the line NA 
would have been the space through which it would have departed from the centre ; therefore the cen¬ 
trifugal force is proportional to NA. Both these forces being then proportional to the same line NA, 
they are equal to each other. 

LEMMA I. 

Quantities and the ratios of quantities, which in any finite time, tend continually to 
equality, and, before the end of that time, approach nearer to each other than by any 
given difference, become ultimately equal. 

If you deny it, let them be ultimately unequal; and let their ultimate difference be D. Therefore 
they cannot approach nearer to equality than by that given difference D ; which is contrary to the suppo¬ 
sition. If a straight and a curve line, continually diminishing, perpetually approach toward equality, 
and at the end of any finite time would vanish together, at the instant in which they are vanishing they 
are equal. 

LEM. II. If in any figure A a c E, terminated by the right lines A a, AE, and the 
curve a c E, there are inscribed any number of parallelograms A b, B c, C d, contained 
under equal bases AB, BC, CD, &c. and the sides, B b, C c, D d, &c. parallel to A a. 
the side of the figure ; and the parallelograms a Ab 1 , b L c m, c M d n, &?c. are complet¬ 
ed ; then, if the breadth of these parallelograms be diminished, and their number augment¬ 
ed continually, the ultimate ratios, which the inscribed figure AK b E c M d D, the cir¬ 
cumscribed figure JalbmcndoE, and the curvilinear figure A a b c d E, have to each 
other, are ratios of equality. 

For the difference of the inscribed and circumscribed figure is the sum of the parallelograms K Z, L in, 
M », D o, that is, (because of the equality of all their bases) the rectangle under one of their bases K b, 
and the sum of their altitudes Aa ; that is, the rectangle AB la. But this rectangle, because its breadth 
AB is diminished indefinitely, becomes less than any given rectangle. Therefore (by Lem. 1.) the in¬ 
scribed and circumscribed, and much more the intermediate curvilinear figure, become ultimately 
equal. 

LEM. III. The same ultimate ratios are also ratios of equality, when the breadths 
AB, BC, CD, £sV. of the parallelograms are unequal, and are all diminished 
indefinitely. 

For, let AF be equal to the greatest breadth ; and let the parallelogram FA af be completed. This 
will be greater than the difference of the inscribed and circumscribed figures ; but, because its breadth 
AF is diminished indefinitely, it will become less than any given rectangle. 

Cor. Hence the ultimate sum of the evanescent parallelograms coincides in every part with the cur¬ 
vilinear figure. Much more does the rectilinear figure, which is comprehended under the chords of the 
evanescent arcs a b, b c, c d , &.c. ultimately coincide with the curvilinear figure. As also the circumscrib¬ 
ed rectilinear figure, which is comprehended under the tangents of the same arcs. And therefore, these 
ultimate figures (as to their perimeter «cE) are not rectilinear, but curvilinear limits of rectilinear 
figures. 

PROP. LXXI. The plane of an orbit in which a body revolves passes through the 
line of projection, and through the centre toward which the centripetal force is directed. 


Chap. VII. 


OF CENTRAL FORCES. 


41 


Let ABCF be the orbit in which the body revolves; S the centre, or point toward which the cen- p ' ate 
tripetal force is directed; and AV the line of projection ; the plane of the orbit will pass through AV * ‘S' 2 
and S ; or the orbit lies in the same plane, with the lines AV, B rf, &c. (lines in which the projectile 
force acts in different parts of the orbit) and with the centre S. 

For, let ABCD represent a part of the orbit described by a body impelled toward S. The body be¬ 
ginning to move by the projectile force from A in the direction ABV would, by that force alone, be 
carried on uniformly in that direction. Suppose the centripetal force to act upon it by separate im¬ 
pulses after equal intervals of time, and that, when the body is carried by the projectile force to B, it 
receives one impulse from the centripetal force, drawing it out of its course toward S, so that by the 
action of both forces together at B, it will (by Prop. XIV.) be made to describe BC in the same time in 
which the projectile force alone would have made it describe B c. The same will take place after equal in¬ 
tervals atC and D. At B, the projectile force is in the direction BV, the centripetal force in the direction 
BS. Let B c and BG, taken in the direction of these forces, represent their ratio to each other, and the pro¬ 
jectile force be to the centripetal as B c to BG. The body (by Prop. XIV.) will describe BC, the diagonal 
of a parallelogram of which B c and BG are the sides. But (El. XI. 1. and 2.) BC is in the same plane 
with B c and BG, that is, with AV, the line of projection, and with BS, in which is the centre or point 
S. The same may be proved concerning the lines CD, and DE. And if the centripetal force act con¬ 
tinually, and not by interrupted impulses, the diagonals AB, BC, &c. will be diminished indefinitely, and 
the ultimate perimeter ADF (by Lem. III. Cor.) will become a curve line, which, from what has bee.n 
shown, must be always in the same plane with the line of projection, and with the centre. 

PROP. LXXII. A body revolving in an orbit describes, by a radius drawn to the 
point toward which the centripetal force acts, equal areas in equal times, and in un¬ 
equal times areas proportional to the times. 

Let ABCD be part of an orbit described by a body which revolves round the point S, toward which Plate 4. 
it is impelled by a centripetal force. If this force be supposed to act upon the body by separate im- Fig. 2. 
pulses, as at B, C, D, when the body receives the impulse at B, it will be drawn out of its course to¬ 
ward S, and (by Prop. XIV.) will describe the diagonal BC in the same time in which the projectile 
force alone would have made it describe B c. After equal intervals, the same will take place at 
C, and at D. 

Since AB, BC, &c. are the lines described in equal times by the body, the areas described round S by 
a radius drawn from the body to S, are ASB, BSC, &c. Now AB, B c, expressing spaces passed over in 
equal times by the uniform motion of the body acted upon by the projectile force alone, are equal bases 
of the triangles ASB, BS c, which, being terminated by the same point S, are of the same altitude ; these 
triangles are therefore (El. I. 38.) equal. And, because the body at Bis by the joint action of the projec¬ 
tile and centripetal forces carried forward in the diagonal BC, of the parallelogram G c, the opposite sides 
thereof, GB, C c, are parallel; and C c is parallel to BS. But BS is the common base of the two triangles 
BSC, BS c. Therefore these triangles, being upon the same base, and between the same parallels, (El. 1. 

37.) are equal. Consequently, ASB, which has been proved equal to BS c, is likewise equal to BSC ; that 
is, the areas described in equal times are equal. And, by composition, any sums of these areas ASC, ASE, 
are to each other as the times in which they are described ; that is, universally, the areas are as the 
times. 

Let the number of these triangles be augmented, and their breadth diminished indefinitely, and (by 
Lem. III. Cor.) their ultimate perimeter will be a curve line ; and therefore the centripetal force will 
act continually; and the above reasoning still being applicable to those triangles whose breadth is in¬ 
definitely diminished, the areas will be as the times. 

Cor. 1. The velocity of a body revolving freely about an immoveable centre is inversely as a per¬ 
pendicular let fall from that centre on a right line that touches the orbit. For since the lines AB, BC, 
are described in equal times, the velocities will be as other lines, which being (by this Prop.) the 
bases of equal triangles, must be inversely as the heights of the triangles; therelore the velocities are 
inversely as these heights, which are measured by perpendiculars let fall from the common vertex, the 
centre S, to the bases, or the bases produced ; that is, when AB, BC, &c. are indefinitely small, to a 
tangent to the orbit. 

Cor. 2. If the chords AB, BC, of two arcs successively described in equal times by the same body 
moving freely, are completed into a parallelogram ABCG, and the diagonal BG, in the position which 
it ultimately acquires when these arcs are diminished indefinitely, be produced toward S, it will pass 
through S, the centre of the centripetal force ; for BG is an indefinitely small part ot the radius SB. 

Cor. 3. If the chords AB, BC, and DE, EF, of arcs described in equal times be completed into the 

6 


42 


OF MECHANICS. 


Book. II. 


Plate 4. 
Fig. 2. 


Plate 4. 
Fig. 1. 


Plate 3. 
Fig. 17. 


parallelograms ABCG, DEFZ, the centripetal forces at B and E, will be to each other in the ultimate 
ratio of the diagonals BG, EZ, when those arcs are indefinitely diminished. For, the motions ot the 
body BC. EF, (by Prop. XVI.) are compounded of the motions B c, BG, and Ef EZ ; but BG and EZ 
are equal to C c and F f which, as appears from this proposition, are generated by the impulses of the 
centripetal force in B and E, and are therefore proportional to those impulses. 

Cor. 4. The forces with which bodies are drawn into curvilinear orbits are to each other as the 
versed sines, -1GB, of the indefinitely small arcs AC, DF, described in equal times, which versed 

sines converge to the centre S, and bisect the chords when those arcs are diminished indefinitely; tor 
such versed sines are half the diagonals of a parallelogram BG, EZ, being bisected by the diagonals 
AC, FD. 

PROP. LXXIII, When a body describes equal areas in equal times about an im- 
moveable point, or proportional areas in unequal times, it is impelled towards that 
point by the centripetal force which retains it in its orbit. 

Let AB, BC, kc. be lines described by the revolving body in equal times; and it may be proved as 
before, that the triangles ASB and BS c being of the same height, and having equal bases AB. B c, are 
equal. But, by supposition, ASB, BSC, are equal; therefore BS c and BSC are equal. And these equal 
triangles, being upon the same base SB, (El. 1. 39.) are between the same parallels ; therefore C c and 
BS are parallel. Because the body at B is acted upon by two forces, the projectile force in the line Be, 
and the centripetal force, and by supposition, these two forces together make it describe BC ; BC is the 
diagonal of a parallelogram of which B c is one side, and the direction of the centripetal force at B is in 
the other side. jYow, in the parallelogram whose diagonal is BC, and one of its sides B c, C c must be 
another: whence the opposite side, that is, the direction of the centripetal force, must be parallel to 
C c; but C c and BS have been proved to be parallel, when equal areas are described in equal times 
about the point S. Therefore, on this supposition, the body at B is acted upon by the centripetal force 
in the direction BS. The same may be shown at every other point C, D, kc. Therefore the centri¬ 
petal force tends to that point round which the body, by a radius drawn thither, describes equal areas 
in equal times. 

LEM. IV. If any arc ACB, of a finite curvature, is subtended by its chord AB, and 
a straight line AD produced both ways touch the arc in the point A, the arc, the chord, 
and the tangent, in their idtimate vanishing state, will be equal. 

The arc being supposed of a finite curvature, or such as may be measured by a circle of a finite di¬ 
ameter ; let BAC be the circle of the curvature ; draw the line AC in that circle parallel to the sub¬ 
tense BD, and complete the triangle BAC. Because the angle DAB (El. III. 32.) is equal to the angle 
ACB, and the alternate angles CAB, ABD, are equal, the triangles, CAB. DAB, are similar. Hence, 
(El. \ I. 4.) AB is to AD, as AC is to BC. The point B approaching continually to A, let BA become 
less than any assignable quantity; then the finite lines AC, BC, approach nearer to the ratio of equality, 
than by any given distance ; therefore likewise AB, AD, which are proportional to AC, BC, and much 
more the intermediate arc, are ultimately equal. 

LEM. V. The nascent or evanescent subtense of the angle of contact, in circles and 
in all curves which have a finite curvature, is as the square of the conterminous arc. 

Let AD, in the semicircle ADC, be the given arc, AB the tangent, and the angle BAD the angle of 
contact. Draw BD, HG, parallel to AC, the diameter; these lines, subtending the angle BAD. are 
called the subtenses of the angle of contact. The arcs AD and AG, having the common term or limit, 
the point A, are called conterminous arcs. Draw the lines DC, GC. If these lines be conceived to 
turn round upon the point C as a centre, so that the two points D, G, and with them the two subtenses 
BD, HG, may approach toward A ; it is manifest, that as these subtenses come nearer to A, they will 
diminish, and at last will vanish in A. At the instant of their vanishing, BD will be to HG. as the 
square of the arc AD is to the square of the arc AG. 

Let ED be drawn parallel to AB, and FG to AH. Then, because AB is a tangent at the point A, 
and consequently (El. III. 18.) perpendicular to the diameter AC, ED, which is parallel to AB. is like¬ 
wise perpendicular to AC; for the same reason FG is perpendicular to AC. And ADC. AGC (El. III. 
31.) are right angles. Therefore (El. VI. 8.) AED is similar to ADC, and AE is to AD. as AD to AC. 
Therefore (El. VI. 17.) the rectangle of AE, AC, is equal to the square of AD. But AE is equal to BD ; 
therefore the rectangle BD, AC, is equal to the square of AD. For the same reason, the rectangle of 
HG, AC, is equal to the square of the chord AG. Consequently, the square of the chord AD is to the 


Chap. VII. 


OF CENTRAL FORCES. 


43 


square of the chord AG, as the rectangle BD, AC, is to the rectangle HG, AC, that is (El. VI. 1 .) as BD 
to HG. But (by Lem. IV.) the arc AD and the chord AD, are ultimately in the ratio of equality, and 
also the arc AG and the chord AG. Therefore the square of the arc AD is to the square of the arc AG, 
at the instant in which they vanish, as BD to HG ; that is, the evanescent subtense of the angle of con¬ 
tact is as the square of the conterminous arc. 

Next; let the subtenses be not parallel to the diameter, but parallel to one another. Let MN, FG, be Fig. 16 
the subtenses parallel to the diameter; and AN, OG, two subtenses parallel to each other, but not to the 
diameter. Because BA is a tangent at the point B, BD (El. Ill, 16.) is perpendicular to BA ; since 
therefore, MN and FG are parallel toBD, they are also perpendicular to BA, and the angles OFG, AMN, 
are equal. But because OG and AN are parallel by construction, the angle FOG (El. I. 29.) is equal to 
the angle MAN. Therefore the triangles FGO, MNA, are similar, and (El. VI. 4 .) AN is to OG, as 
MN is to FG. But it has been proved that MN is ultimately to FG, as the squares of the conterminous 
arcs ; therefore AN is ultimately to OG, as the squares of the conterminous arcs BN, BG. 

Lastly ; suppose both AN and OG directed to C, the centre of the circle. In this case, each of these 
would be a semidiameter, continued from G and N respectively to the tangent BA. In their ultimate ’ 

state these lines AN, OG, must coincide in the point B, and in the same right line BC ; and therefore 
will become parallel, and will be, from what has been shown, ultimately as the squares of the conter¬ 
minous arcs. 

If GC, DC, be beginning to move from A, they are in their nascent state ; and it is manifest that the pig, 17 , 
subtenses in this state are the same, and therefore have the same ratio, as in the evanescent state. 

Cor. 1. Hence, because the tangents AB, AH, the arcs AD, AG, and their sines ED, FG, become 
ultimately equal (by Lem. IV.) to the chords AD, AG, their squares also will be ultimately as the sub¬ 
tenses BD, HG. 

Cor. 2. The same squares are also ultimately as those versed sines of the arcs, which bisect the 
chords and converge to a given point. For by the second case of this proposition these versed sines AE, 

AF, are as the subtenses BD, HG, or AN, OG. 

Cor. 3. Hence these versed sines AE, AF, are as the squares of the times in which a body describes 
the arcs AD, AG, with given velocities. For the spaces AD, AG, described with given velocities, are 
as the times, and the squares of the spaces as the squares of the times; but (by last Cor.) the squares of 
these spaces are as the versed sines AE, AF; therefore these versed sines are as the squares of the 
times in which the arcs AD, AG, are described. 

LEM. VI. The nascent or evanescent subtense of the angle of contact is equal to the 
square of the conterminous arc divided by the diameter. 

It has been shown in the preceding Lemma, that BD is to AD, as AD is to AC. Therefore BD Plate 3 . 

AD s Fie 17 

multiplied into AC is equal to the square of AD, and BD = - - . But (by Lem. IV.) the arc AD is S 

AL 

ultimately equal to the chord AD ; therefore the nascent or evanescent subtense BD is equal to the square 
of the arc AD divided by the diameter AC. 

PROP. LXXIV. The centripetal forces of bodies, revolving in different circular 
orbits about the same centre toward which they tend, are as the squares of the arcs 
described in the same time, divided by the radii of the circles. 

In the circular orbits BND, RLE, let bodies revolve about the centre C, toward which they tend. Plate 3 
Let them in the same time describe the indefinitely small arcs BG, RL. Then because the projectile Fig. 16. 
forces would carry them in the same time through the tangents BF, RH, and the spaces through which, 
at the points G and L, they have been drawn from the tangents toward the centre by the centripetal 
force, are FG, HL; the centripetal forces must be as FG and HL. And (by Lem. VI.) the evanescent, 
or nascent, subtense FG is equal to the square of the arc BG divided by BD, and the evanescent, or 
nascent, subtense HL is equal to the square of the arc RL divided by RE. Therefore the subtense FG 
is to the subtense HL as the square of the arc BG divided by BD or its half BC, is to the square of the 
arc RL, divided by RE, or its half RC. Therefore the centripetal forces, when the arcs are nascent, are 
in the same ratio ; that is, as the squares of the arcs divided by the radii. 

And this is true, whatever arcs BG and RL be taken, if they be described in the same time ; for the 
nascent arcs will be as the velocities; and any other arcs BND, RLE, described in any given time, 
will be also as the velocities; therefore, the arcs BND, RLE, are as the nascent arcs BG, RL, and 
their squares are likewise proportional. But the centrifugal forces are as the squares of the nas¬ 
cent arcs, BG, RL, divided by the radii BC, RC; therefore these forces are as the squares of any other 
arcs, BND, RLE, divided bv the radii of their circles. 



41 


OF MECHANICS. 


Book II. 


Plate 3. 
Fig. 16. 


PROP. LXXV. The centripetal forces of equal bodies revolving in circular orbits, 
are as the squares of the velocities directly, and the radii of the orbits inversely. 

Because arcs described in the same time are as the velocities, and that the centripetal forces are 
(by Prop. LXXIV.) as the squares of the ai*cs described in the same time divided by the radii, these 
forces are also as the squares of the velocities divided by the radii, that is, as the squares of the veloci¬ 
ties directly, and the radii of the orbits inversely. 

Cor. Hence the centripetal forces of equal bodies, at equal distances from the centre, are as the 
squares of the number of revolutions in any given time; for this number is as the velocity with which 
the body moves. 

PROP. LXXVI. The centripetal forces of equal bodies revolving in equal circular 
orbits are inversely as the squares of their periodical times. 

The circular orbits or spaces being equal, the times in which these are described, or the periodical 
times , are (by Prop. V.) inversely as the velocities; and therefore the squares of the periodical times 
are inversely as the squares of the velocities, or the squares of the velocities are inversely as the 
squares of the periodical times; but (by Prop. LXXV.) the centripetal forces are as the squares of the 
velocities; therefore these forces are inversely as the squares of the periodical times. 


PROP. LXXVII. The centripetal forces of equal bodies revolving in unequal cir¬ 
cular orbits, if the periodical times are equal, are as the radii of the circles. 


Let one body revolve in the circular orbit BND, and another, in the same time, in the circular 
orbit RLE. Because the periodical times are equal, each body in any given part of its periodical time 
will describe an equal number of degrees in its respective orbit, that is, will describe similar arcs. 
The arcs BN, RL, being similar, will be described in equal portions of the periodical time ; therefore 
(by Prop. LXXIV.) the centripetal forces will be as the squares of the similar arcs BN, RL, divided by 

BN 2 RL* 

the radii BC, RC ; that is, as - - - , - to ——-. But because similar arcs are to each other as the circum- 

Dt KL 


ferences, or radii, of circles, BN is to RL as BC to RC, and consequently, BN* to RL 2 as BC 2 to RC 3 . 
BN 2 RL 2 BC 2 RC 4 

Therefore -gg- is to as gg- is to — ; that is, as BC to RC. But the centripetal forces (Prop. 
BN* RL* 

LXXIV.) are as -gg- to -gg ; therefore these forces are as BC to RC; that is, as the radii of the orbits 
in which the bodies move. 


PROP. LXXVIII. The centripetal forces of equal bodies revolving in circular 
orbits, are as the radii of the orbits directly, and the squares of the periodical times 
inversely. 

If the periodical times are equal, and the radii unequal, the forces are (by Prop. LXXVII.) as the 
radii. If the radii are equal, and the periodical times unequal, the forces by (Prop. LXXVI.) are in¬ 
versely as the squares of the periodical times. Therefore, if both the radii and periodical times are 
unequal, the forces will be in the compound ratio of both, or as the radii directly, and the squares of 
the periodical times inversely. 


PROP. LXXIX. When bodies revolve round the same centre, if the squares of 
their periodical times are as the cubes of their distances from the centre, the centripe¬ 
tal forces will be inversely as the squares of their distances. 


Let the distances of the two bodies be expressed by D, d ; and the periodical times by P » • then 
by the supposition, P 2 : jo* :: D 3 : d 3 . ’ r ' ’ 

By Prop. LXXVIII. the centripetal forces are as the distances directly, and the squares of the peri¬ 


odical times inversely; that is, (taking C, c, for the centripetal forces) C : c - and by suppo- 


PI pA 

D d 


sition P 3 : p* :: D 3 : d 3 ; therefore, substituting D 3 , d*, for P*, p 2 , C : c : : — ; . that is, 

D 3 d 3 } ’ 


:: D*''d 2 ’ 


i 


C : c 



Chap. VII. 


OF CENTRAL FORCES. 


45 


is to 


1_ 

d* 


m- 


tliat is, the centripetal forces are inversely as the 


and, because where the dividend is given, the quotient is inversely as the divisor, -in¬ 
versely as D 1 to d 2 . Therefore C : c :: d 2 : D 2 
squares of the distances. 

Schol. 1. Let C, c, express the central forces; A, a , the arcs described; V, v, the velocities with 
which the bodies move ; P, p, the periodical times of their revolutions; D, d , the radii or distance 
fiom the centre; and N, the number of revolutions in a given time j the preceding Propositions mav 
be thus expressed. 

The bodies being equal, 


Prop. LXXIV. 

C : 

A 2 

C ” R 

a 2 

r 

LXXV. 

C : 

V» 

v 2 



C :: D 

d 

Cor. 

C : 

c :: N l 

: n 2 . 

LXXVI. 

C : 

1 

C ^ P 2 

1 

: —- or 
v 2 

LXXVII. 

C : 

c :: D : 

: d. 

LXXVIII. 

C : 

D 

C :: P^~ 

d 

P 2 ' 

LXXIX. 

IfP 1 :p 2 

::D 3 : d 3 , 

C : c :: 


ttt : —tt or r> % : P : 


1 _ 

D : 


or d 1 : D*. 


Schol. 2 . Since it was proved (Prop. LXX.) that the centripetal and centrifugal forces are, in circu¬ 
lar orbits, equal to one another, the preceding Propositions, being demonstrated respecting the centrip¬ 
etal force, are also true of the centrifugal force; and it may be asserted universally, that the central 
forces are in the ratios above expressed. 

These propositions may be confirmed by the following experiments, on the whirling tables. 

Exp. 1 . Let two equal balls be placed at equal distances from the centre of motion on the whirling 
tables ; and let one table revolve twice whilst the other revolves once ; the ball on the table whose 
number of revolutions is, with respect to that of the other in the same time, as 2 to 1 (or the periodical 
times as 1 to 2) will raise 4 times the weight raised by the other ball; that is, (according to Prop. 
LXXV. and Cor.) the radii being equal, C : c:: V* : v* : : N 3 : n 2 ; or (by Prop. LXXVI.) :: p 2 : P*. 

2 . Let two equal balls be placed on tables whose number of revolutions in the same time is as 2 to 
1 ; let the ball on the table, whose number of revolutions is 2, be placed at half the distance from 
the centre, at which the ball on the table, whose number of revolutions is 1, is placed ; whence their 
velocities will be equal. The ball at the distance 1, will raise double the weight raised by the ball at 
the distance 2; that is, according to Prop. LXXV. the velocities being equal, C : c : : d : D. 

3. Let two equal balls revolve on tables whose periodical times are equal; and let the distances of 
the balls from the centre be to each other as 2 to 1 ; the ball which is at the distance 2 will raise 
double the weight raised by the ball which is at the distance 1; that is, according to Prop. LXXVII. 
C : c :: D : d. 

4. Let equal balls be placed on tables whose periodical times are as 2 to 1 ; let the ball on the 

table whose periodical time is 2, be placed twice as far from the centre as the ball whose periodical 
time is 1 ; the ball whose distance is 2, and periodical time 2, will raise half the weight raised by 
the ball whose distance is 1, and periodical time 1 ; that is, according to Prop. LXXVIII. 
p # D 2 1 

; c:: P 2 : p* :: 7 : 2- 

5. Let the equal balls be so placed on different tables, that the distance of one from the centre may 
be to that of the other as 2 to 3|-; let that ball which is at the least distance revolve twice in the same 
time in which the other ball revolves once; the periodical time of the ball at the less distance, is to 
that of the ball at the greater, as 1 is to 2, and the square of the periodical times will be as 1 to 4, and 
the cubes of the distances are 8, and 31.75 ; but 1 : 4 :: 8 : 32, therefore the squares of the periodical 
times being in this case nearly as the cubes of the distances, the weight raised by the ball whose dis¬ 
tance is 2, will be to that raised by the ball whose distance is 3£, as the square of 3|- is to the square of 
2; that is, nearly as 10 to 4, or 5 to 2. 

PROP. LXXX. The centrifugal forces of revolving bodies are as their quantities of 
matter. 

For the whole centrifugal force of any body is made up of the centrifugal forces of each particle of 


46 

OF MECHANICS. Book II. 


matter of which it consists; and therefore the more numerous the particles of matter in any body are. 
the greater will be its centrifugal force. 

Exp. Let two glass tubes be half filled with water; into one put some leaden shot, and into the 
other a few small round pieces of light w ood ; let the orifice of each tube be closed by a cork ; lasten 
the tubes to an inclined plane, and let the low r er end of it rest upon the centre of a whirling table. On 
turning the table, the bodies will be carried by their centritugal forces from the centre; and the heavier 
bodies will recede farther from the centre than the lighter. See Ferguson's Lectures. 

Cor. Hence, when the revolving bodies are not equal, the centritugal lorces are in the ratios laid 
down in the preceding propositions multiplied into their quantities ot matter. Thus Q, < 7 , expressing 
the quantities of matter, and the other expressions remaining as in Prop. LXXIX. Schol 

C : c :: Q : q. 

QV 2 qv 3 

G ‘ C " D ‘ d 

C : c : :QN* : q n* 

C . c . — • -2- 

U P 2 ' p* 

C c :: QD : qd 

QD qd 

c : c fT : pr 

Cor. Hence the central forces will be equal, whenever the expressions proportional to them are 
equal; thus, C = c if QD = qd. 

Any of the above proportions may be confirmed by experiment; for example ; 

Exp. 1 . Let the two balls A, B, be as 2 to 1 ; let the distance of the ball A be to that of the ball B 
from the centre, as 2 to 1 , and the periodical time of the ball A be twice that of the ball B ; their ve- 

Q q 

locities will be equal; therefore the centrifugal force of A w ill be to that of B, as — is to that is, as 

1 to 1, or A and B will raise equal weights. 

2, 3. Let the same balls revolve about a fixed point, and have their distances reciprocally propor¬ 
tional to their quantities of matter, their centrifugal forces (compare Prop. LXXV. and LXXX.) will be 
equal, and they will balance each other. This may be shown by tw o balls suspended freely and united 
by a cord, having the point of the cord which is directly above the centre of the table at distances from 
the balls reciprocally as their weight; or by two balls united by a wire, and resting in equilibrio on a 
forked support fixed in the centre of the tables, which will continue in equilibrio w hen the tables are 
turned. 

In like manner other cases may be confirmed by experiment. 

LEM. VII. If a body revolves freely in any orbit about an immoveable centre, and 
in an indefinitely small time describes any nascent arc ; and the versed sine of the arc be 
drawn which may bisect the chord , and being produced may pass through the centre of 
force ; the centripetal force, in the middle of this arc, will be as the versed sine directly 
and the square of the time inversely. 

Plate 4. 
Fig. 3. 

Let two bodies revolve round their centre of force S, s ; let QPM, qpm , be the nascent arcs de¬ 
scribed in any times, T, t; and let PB, p 6 , or QR, A 0 , be the versed sines bisecting the chords, antt 
when produced, passing through S the centre of force. Supposing the arcs QPM, ApN, to be describ¬ 
ed in the same time with different forces C, c ; by Prop. LXXII. Cor. 4. QR : A a :: C : c. Hence, 
supposing the forces to be equal, QR is equal to A a described in the same time; and (by Lem. V.) 
QR or A a : qr :: Ap* : qp 3 ; that is, since the motion in the arcs is uniform, A a : qr :: T 2 t 2 . 
Therefore, supposing both the times and forces different, and compounding these ratios, QR : qr :: C x 

T x : c x t 2 ; whence C : c :: : ILL, 

Plate 4. 
Fig. 4, 

Cor. 1 . If a body P, revolving about the centre S, describe a curve line APQ, and a right line 
ZPR touch that curve in any point P ; and, from any other point Q of the curve, QR be drawn par¬ 
allel to the distance SP, meeting the tangent in R ; and QT be drawn perpendicular to the distance SP ; 

SP 2 x QT 3 

the centripetal force will be reciprocally as the quantity -—-, if this be taken of that magnitude 

Vc^iV 





Chap. VII. 


OF CENTRAL FORCES. 


47 


which it ultimately acquires, supposing the points P and Q continually to approach to each other. For 
QR is equal to the versed sine of double the arc QP, in whose middle is P; and double the triangle 
SQP, or SP x QT, is proportional to the time, in which that double arc is described (by Prop. LXXII.) 

Q R Q T 

and therefore may be used for the exponent of the time. Whence C : c :: • -■- 


SP 2 x QT 2 s P 2 x qt* 


QR 


q r 


SP* X QT a ' sp* Xqt* 1 
; or the centripetal forces are reciprocally as 


that is, C is to c reciprocally as 
SP’ x QT 2 
QR ‘ 

Cor. 2. Hence, if any curvilinear figure APQ is given ; and therein a point S is also given, to which a 
centripetal force is perpetually directed; the law of centripetal force may be found, by which the body 
P, continually drawn back from a rectilinear course, will be retained in the. perimeter of that figure, 

SP* x QT 2 

and will describe the same by a perpetual revolution. That is, we are to find the quantity 


reciprocally proportional to this force. 


QR 


PROP. LXXXI. If equal bodies, revolving in ellipses, describe equal areas in equal 
times, their centripetal forces are to one another inversely as the squares of their dis¬ 
tances from the foci of the ellipse toward which they tend. 


Let S be the focus; let a body P, tending toward S, describe a part of the ellipse PQ; join SP; Plate 4. 
draw QR to the tangent YZ, parallel to SP; join PC, and produce it to G. Complete the parallelogram Fl S- 4 - 
QxPR, produce Qx to v, Qi> is ordinately applied to GP; draw DK, a diameter parallel to YZ, and 
draw IH from the other focus H to SP parallel to YZ; join HP, and draw QT perpendicular to SP, as 
also PF to DR. 

EP is equal to the greater semiaxis AC. For, because CS is equal to CH, ES is equal to El, (El. 

VI. 2 .) whence EP is half the sum of PS, PI ; that is, of PS, PH, for (Simson’s Conic Sect. II. 11. Cor.) 
the angle IPR is equal to HPZ ; whence (El. I. 29.) the angle PIH is equal to PHI, and PI is equal to 
PH; and PS, PH, together, (Simson’s Conic Sect. II. 1 .) are equal to the whole axis 2AC. EP there¬ 
fore is equal to AC. 

2BC* 

Putting L for the principal lotus rectum of the ellipse, L (by definition) is equal to — 77 ,— (for AC -. 

AL 


T 2CR 2 

CB :: CB : —, whence ~ ■ = L.) And L X QR : L X P v : : QR : P v ; and QR = Pat; and P x : 

P v :: PE : PC; whence L X QR : L X P v : : PE or AC : PC. And (El. VI. 1.) L X Pao : Gn X 
P v :: L : G v ; and (Sims. II. 15.) G v X P® : Q» 2 :: PC 2 : DC 2 . And (Lem. IV.) the points Q and 
P continually approaching, Qo> 2 is to Q x 2 ultimately in the ratio of equality. And (since the triangles 
QT x, EPF, are similar, for Q x T = PEF, and QT x to EFP) Q x* or Q t>* : QT 2 :: EP’ or AC 2 : PF 2 . 
But because (Sims. II. 20 . Compare Vince’s Con. Sect. II. 10. Cor. 1 .) parallelograms about conjugate 
diameters are equal to the rectangle under the axes, the rectangle PF, DC, is equal to the rectangle 
ACB, whence PF : AC :: CB : DC, and AC 2 : PF 2 : : CD* : CB», wherefore Q v* : QT’ :: CD 2 : CB’, 
Compounding the following ratios, 

L X QR : L X Fv :: AC : PC, 

L xP^ : Gtu X P® :: L : Gd, 

% G^ X Pd: Qx* :: PC* : CD* ; 

Qv 2 : QT’ :: CD* : CB»; 

And, striking out the equal quantities, L x QR : QT 3 :: AC X L X PC :G® x CB*. 

Then substitute for AC X L its equal 2CB 2 , and 

L X QR : QT* : : 2 BC* X PC :Gd X BC* 
or BC 2 x 2PC : Gv X BC’ 
or 2 PC : Gv. 

But the points Q and P continually approaching without end, 2 PC and G v are equal; wheretore 

SP’ 

L x QR and QT’, proportional to these, are also equal. Multiply these equals into an d L X SI 32 


SP* x QT 2 
will become equal to -——-. 

Therefore (by Lem. VII. Cor. 1 and 2.) the centripetal force is reciprocally as L X SP*; that is, 
since L is a given quantity, as SP 2 , or in a duplicate ratio of the distance SP. 











BOOK III 


Plate 5. 
Fig. 2. 


OF HYDROSTATICS AND PNEUMATICS; 

OR THE LAWS OF 

INCOMPRESSIBLE AND COMPRESSIBLE FLUIDS. 


PART I. 

OF HYDROSTATICS. 

CHAPTER I. 

Of the Weight and Pressure o f Fluids. 

Def. I. A FLUID is a body, the parts of which yield to any force impressed upon 
them, and easily move out of their places. 

PROPOSITION I. 

The weight of fluids is as their quantities of matter. 

Since each particle of any fluid gravitates toward the earth, the greater is the number of particles, 
that is, the greater the quantity of matter in any mass of fluid, the greater will be the weight of that 
mass. 

Exp. 1. The different pressures of different columns of fluid in the same vessel at different depths, 
appear from the different quantities of fluid discharged, at diffei’ent depths, in the same time from orifices 
of the same bore. 

2. If the air be exhausted from a tube in part filled with water, and the tube be closed up, the solid¬ 
ity of the particles of water will be perceived by the sound produced by suddenly lifting up the tube. 

Cor. Fluids gravitate in fluids of the same kind. For they cannot lose the property of gravity 
which belongs to all bodies By such a change of situation. 

Exp. Suspend a stopped phial from one arm of a balance, in a vessel of water, and balance it by 
weights from the opposite arm of the balance ; upon unstopping the phial under water, a quantity of 
water will rush into it, by which the weight will be increased as much as the weight of water in the 
phial. 

PROP. II. When the surface of a fluid is level, the whole mass will be at rest. 

Let ABCD be a vessel containing water, the level surface of which is EF. Conceive the whole 
mass of fluid in the vessel to be divided into thin strata , or plates, RS, TV, XY, &c. lying horizontally 
one above another; and into small perpendicular columns GH, IK, LM, &c. contiguous to each other. 
In the stratum XY, and the columns IK, LM, let m, n, be two adjacent drops. Neither of these drops 
can move toward the column in which the other is, without driving that other out of its place, because the 
fluid is supposed incompressible. But, with whatever force the particle m endeavours to .displace the 
particle n, this force is counterbalanced by an equal and contrary effort on the part of n ; because 
(Prop. I.) they are equally pressed by the equal columns above them ; consequently the particles will 
be at rest. 

PROP. III. Any part of a fluid at rest presses, and is pressed, equally in all 
directions. 




Chap. I. 


THE PRESSURE OF FLUIDS. 


49 


For (Def. I.) each particle is disposed to give way on the slightest difference of pressure ; and, by 
supposition, each particle is pressed by the contiguous particles in such manner as to be kept at rest 
in its place ; it is therefore pressed with an equal degree of force on all sides; and, consequently, (Book 
II. Prop. III.) it presses equally in all directions. 

Cor. Hence the lateral pressure of a fluid is equal to the perpendicular pressure. This is one of 
the most extraordinary properties of fluids, and can be conceived to arise only from the extreme facility 
with which the component particles move among one another. 

Exp. 1 . Into several tubes, bent near their lower ends in various angles, pour a sufficient quantity of 
mercury to fill the lower parts of their orifices ; then dip them into a deep glass vessel filled with water, 
keeping the orifice of the longer legs above the surface; whilst the tubes are descending, the mercury 
will be gradually pressed upward in the tubes, and the pressure will be equal at any given depth, what¬ 
ever be the direction of the pressing column of fluid in the shorter leg of the tube. Oil may be used 
instead of mercury. 

2 . Dip an open end of a tube, having a very narrow bore, into a vessel of quicksilver; then, stop¬ 
ping the upper orifice with the finger, lift up the tube out of the vessel; a short column of quicksilver 
will hang in the lower end, which, when dipped in water lower than 14 times its own length, will, 
upon removing the finger, be suspended and pressed upward. 

3. Let a large open tube be covered at one end with a piece of bladder drawn tight; pour into the 
tube a quantity of coloured water sufficient to press the bladder in a convex form ; then, dip the covered 
end of the tube slowly into a deep vessel of water; the bladder, by the upward pressure will become 
first less convex, then plane, and at last concave. 

4 . If the like be done with several tubes, whose covered orifices are cut obliquely at different angles, 
the lateral pressure will be seen to increase with the depths to which the tubes are immersed. 

5. Let a circular piece of brass, whose upper surface is covered with wet leather, be held close to 
one orifice of a large open tube, by means of a cord or wire fastened to the middle of the plate, and 
passing through the tube ; let the plate, thus kept close to the orifice of the tube, be immersed with 
the tube into a large vessel of water; when the plate is at a greater depth than 8 times its thickness 
in the water, the cord or wire may be left at liberty, and the upward pressure of the fluid will keep the 
plate close to the tube. 

6 . Let a small bladder, tied closely about one end of an open tube, having a large bore, be filled 
with coloured water till the water rises above the neck of the bladder; upon immersing the bladder 
into a vessel of water, the bladder will be compressed on all sides, and the coloured water will be raised 
up in the tube in proportion to the depth to which the bladder is sunk. 

PROP. IV. When a fluid flows through a tube which is wider in some parts than 
in others, the velocity of the fluid will, in every section of the tube, be inversely as the 
area of the section. 

Let ADMN, a bended tube larger at IL than at FG, be filled with water to the height ADFG. Let Plate 5. 
the water be forced downward in the part ADBP, and consequently be made to rise in the other part Fi g- L 
KHMN. It is manifest, that the water which is forced out of one part of the tube, is driven into the 
other. Hence equal quantities pass through every section of the tube at the same time; for if less, or 
more water passed through the section FG than through 1L in the same time, the quantity of water 
between FG and IL must be increased or diminished, which cannot be, since no cause is supposed which 
could increase or diminish it. But if equal quantities pass through unequal parts of the tube in the 
same time, the wafer must run proportionally faster where the tube is narrower, and slower where it 
is wider. If, for example, as much water runs through the section FG, as runs in the same time through 
the section IL, the water must move as much faster at FG than it moves at IL, as the tube is narrower 
at FG than at IL; that is, the velocity is inversely as the area of the section. 

Cor. The momentum will be the same in every section of the tube ; for the quantity of water at 
each section is directly as the area of the section, and the velocity is inversely as the area; therefore 
the velocity is inversely as the quantity of matter*; whence (Book II. Prop. XI.) the momentum is every 
where the same. 

Schol. Hence we may account for the suspension of the fluid in a tube, the upper part of whose bore 
is capillary, and the lower of a much larger dimension, as w'as seen in the experiment, Book I. Prop. 

VH. 

Let there be a tube consisting of two parts DR and RCK, of different diameters; DR, the smaller Plate 5. 
part of the tube, is able (Book I. Prop. VIII.) to raise water higher than the other; let then the height F 'g- 3 - 
to which the larger would raise it be TC, and that to which it w ould rise in the lesser, if continued 
down to the surface of the fluid, be XH. If this compound tube be filled with water, and the larger 


.50 
rig. 4. 


Fig. 4. 


Plate 5. 
Fig. 5. 


Plate 5. 
Fig. 1. 


OF HYDROSTATICS. Book III. 

orifice CK be immersed in the same fluid, the surface of the water will sink no farther than XL, the 
height to which the lesser part of the tube would have raised it. But if the tube be inverted, and the 
smaller orifice XL be immersed, the water will run out till the surface falls to TF; the height to which 
the larger part of the tube would have raised it. 

Let the tube Dll be conceived to be continued down to HI; and let it be supposed that the fluids 
contained in the tube XLHI, and the compound one XLKC, are not suspended by the ring of glass at 
XL, but that they press upon their respective bases, HI and CK. Let it farther be supposed that these 
bases are each of them moveable, and that they are raised up or let down with equal velocities; then 
will the velocity with which XL, the uppermost stratum of the fluid XLCK, moves, exceed that of the 
same stratum, considered as the uppermost of the fluid in the tube XLHI, as much as the tube RCK is 
wider than DR, (by this Prop.) that is, as much as the space MNKC exceeds XLIH. Consequently, the 
effect of the attracting ring XL, as it acts upon the fluid contained in the vessel XLCK, exceeds its ef¬ 
fect, as it acts upon that in XLHI, in the same ratio. Since, therefore, it is able to sustain the weight 
of the fluid XLHI by its natural power, it is able, under this mechanical advantage, to sustain the weight 
of as much as would fill the space MNKC ; but the pressure of the fluid XLCK is equal to that weight, 
as having the same base and an equal height (as will be shown by Prop V r I.) Its pressure, therefore, 
or the tendency it has to descend in the tube, is equivalent to the power of the attracting ring XL, for 
which reason it ought to be suspended by it. 

Again, the height at which the attracting ring in the larger part of the tube is able to sustain the 
fluid is no greater than NF, that to which it would have raised it, had the tube been continued down to 
MN. For here the power of the attracting ring acts under a like mechanical disadvantage, and is there¬ 
by diminished, as much as the capacity of the tube TFNM is greater than that of HIXL ; because, if 
the bases of these tubes are supposed to be moved with equal velocities, the rise or fall of the surface of 
the fluid TFXL would be so much less than that of TFMN. And, since the attracting ring TF is able, 
by its natural power, to suspend the fluid only to the height NF in the tube TFMN; it is in this case 
able to sustain no greater pressure than what is equal to the weight of the fluid in the space HIXL; but 
the pressure of the fluid TFXL, which has equal height, and the same base with it, is equal to that 
weight; and therefore is a balance to the attracting power. 

From hence we may clearly see the reason, why a small quantity of water put into a capillary tube, 
which is of a conical form, and laid in a horizontal situation, will run toward the narrow end. For let 
AB be the tube, and CD a column of water contained within it; when the fluid moves, the velocity of 
the end D will be to that of the end C reciprocally as the cavity of the tube at D to that at C, (by this 
Prop.) that is, (El. XII. 2.) reciprocally as the square of the diameter at D, to the square of the diame¬ 
ter at C ; but the attracting ring at D is to that at C, singly as the diameter at D to the diameter at C. 
Now, since the effect of the attraction depends as much upon the velocity of that part of the fluid where 
it acts, as upon its natural force, its effect at D will be greater than at C ; for though the attraction at 
D be less in itself than at C, yet its loss of force upon that account, is more than compensated by the 
mechanical advantage it has arising from hence, that the velocity of the fluid in that part is more in¬ 
creased than the force itself is diminished at D. The fluid will therefore move towards B. See, on 
this subject, Mr. Vince’s Principles of Hydrostatics, p. 65—9. 

PROP. V. In bended cylindrical tubes, fluids at rest will be at the same height on 
each side. 

In the tube ADMN, filled with water to the height AD, the water cannot descend from AD, without 
rising toward MN. The water in each side of the vessel may therefore be considered as two forces 
acting upon each other in contrary directions ; and consequently these two masses of fluid will only be 
at rest when their momenta are equal; that is, (Book II. Prop XI. Cor.) when the quantities of matter 
are inversely as the velocities, or (Prop. IV.) directly as the area of the section through which it flows. 
Thus, at the sections BP, KH, the momenta are equal, when the quantities of matter, or cylindrical 
masses of fluid are as the areas of the sections ; that is, as the bases of the cylinders ADBP, FGHK. 
But cylinders are as their bases (El. XII. 11.) only when their perpendicular heights are equal. , There¬ 
fore the momenta of the two cylinders of fluid will be equal, and consequently the mass will be at rest 
only when the perpendicular heights of each column are equal. 

Exp. 1 . In a bended tube of large but unequal bore, water will rise to the same height on each side. 

2 . Let water spout upward through a pipe, having a small orifice inserted into the bottom of a deep 
vessel; it will rise nearly to the height of the upper surface of the water in the vessel. The resis¬ 
tance of the air, and of the falling drops, prevents it from rising perfectly to the level. 

Cor. If, therefore, a pipe convey a fluid from a reservoir, it can never carry it to a place hi°-her 
than the surface of the fluid in the reservoir. 

Sx’hol. In this demonstration, we do not consider the velocity with which the two columns of fluid 


Chap. I. 


THE PRESSURE OF FLUIDS. 


51 


are moving, but the velocity with which, if they move at all, they must begin to move. And since, if 
their perpendicular height is the same, the velocity with which they must begin to move will be in¬ 
versely as their respective quantities of matter, they cannot begin to move but with equal momenta; 
and their motions must be in contrary directions, because one column cannot descend without making the 
other ascend; therefore those equal momenta would destroy each other. These two columns then, 
making a continual effort to move with equal momenta in contrary directions, counterbalance each other. 

PROP. VI. The pressure of fluids is proportional to the base, and the perpendicu¬ 
lar height of the fluid, whatever be the form of the vessel or quantity of the fluid. 

Case 1. Let the fluid be contained in a perpendicular cylindrical vessel. Plate 5. 

In such a vessel, ABCD, because the whole weight of the fluid, and no other force, presses directly Fig- 2 • 
upon the bottom CD, the pressure (by Prop. I.) must be as the quantity; that is, (El. Xil. 11, 14.) as 
the base and perpendicular height of the fluid. 

Case 2 . Let the fluid be contained in a perpendicular vessel, the bottom of which is equal to that of 
the cylinder in the last case, but its top narrower than the bottom. 

Let the vessel DBLP, have the portions of its base LA, CP, each equal to OR. From Prop. I. and Plate 5. 
III. it appears, that each of these portions are equally pressed by the column DBOR, as the base OR. Fig' 6. 
In like manner, every portion of the base LP equal to OR is as much pressed as OR. Therefore the 
whole base LP is as much pressed as if the vessel was of the cylindrical form FHLP. 

Or thus ; because (by Prop. V.) if a tube were inserted at NT, of the diameter OR, the water, be¬ 
ing at the height DB, would rise to the level FE, there must at NT be an upward pressure toward F 
sufficient to All up the columns of fluid FELA; that is, equal to the weight of as much water as would 
fill the space FENT. Consequently the re-action, that is, the pressure upon ths base LA, must be equal 
to the weight of as much water as would fill FENT. But the base LA supports this re-action, and 
likewise the weight of the water NTLA, which are together equal to the weight of DBOR. The base 
LA, therefore, sustains a pressure equal to the weight of the column DBOR. And every equal portion 
of the base may, in the same manner, be shown to sustain an equal pressure. Therefore, the pressure 
on the base is the same in vessels of the form supposed in this case, as in cylinders of equal bases, and 
of the same altitude with these vessels. The same may be shown with respect to a vessel of the form 
of plate 5, fig. 7. 

Case 3. Let the vessel be of the same base and altitude, but have its top wider than the base. 

Let the fluid of the vessel be divided into strata EF, GH, IK, &,c. Let us 'also imagine the bottom of Plate 6 . 
the vessel C to be moveable, that is, capable of sliding up and down the narrow part of ' f ”om Fig. 8 . 

C to GH. Let it further be supposed that this moveable bottom is drawn up or let d< 
velocity, wffiile the vessel itself is fixed and immoveable; it is evident the lowest strati nich 
tiguous to the bottom, will be raised or let down with the same velocity, and will the re have 
mentum proportional to that velocity, and the quantity of matter it contains ; but (1 p. D Cor.) 
the rest of the strata will have the same momentum ; consequently, the momentum of an 
that is, of the whole fluid, is the same as if the vessel had been no larger in any one part than it ... 
the bottom, for then the momentum of each stratum would also have been as great as that of the lowest. 

The pressure, therefore, or action of the fluid, with which it endeavours to force the bottom out of its 
place, is as the number of strata, that is, the perpendicular height of the fluid, and the magnitude of the 
lowest stratum, that is, the base. 

Case 4. Let the fluid be in an inclined cylindrical vessel. 

In the inclined cylindrical vessel ABNI, as much as the fluid is prevented from pressing upon the pi a te 5. 
base NI, by being in part supported by the side of the vessel AN, so far is the pressure upon the base Fig. 9. 
increased by the re-action of the opposite side BI, which is equal to the action of the former, because 
the fluid, pressing every way alike at the same depth below the surface, exerts an equal force against 
both the sides. The base NI is therefore pressed with the same force with which it would be pressed, 
if the fluid contained in the vessel ABNI was included in the vessel EDIO, having an equal base, and 
the same perpendicular height with the vessel ABNI; that is, (by the first case) the pressure is as the 
base NI and altitude CN. 

Since then, the pressure upon the base of vessels, either wider or narrower at the top than the bot¬ 
tom, and likewise the pressure upon the base of vessels inclined to the horizon, is equal to that upon 
the base of a cylindrical vessel of the same base and height, the sides of which are perpendicular to 
the horizon; and since the pressure upon the base of such a cylinder i*as the base and height; the 
pressure upon the bottom of all vessels filled with fluid is proportional to their base and perpendicular 
height. 

Exr. 1. Let two tubes of different forms be successively applied to the same moveable circular base, 
suspended by a wire, passing from the centre of the. base through the tubes, to the beam of a balance ; 


V 


I 


52 


OF HYDROSTATICS. 


Book III. 


Plate 5. 
Fig. 6. 


Plate 5. 
Fig. 12. 


when the different tubes are filled to the same height, it will require the same %veight at the opposite 
end of the balance to keep the base from sinking, lienee any quantity of fluid, how small soever, may 
be made to balance and support any quantity how great soever, which is called the hydrostatical paradox. 

2 . Let two tubes, the one cylindrical, the other of the form of a speaking trumpet, have their bases 
of equal diameter, covered with bladder, and inserted in a vessel of water, as in Prop. 111. Exp. 3. the 
bladder will become plane at the same depth in both; from whence it appears, that since the upward 
pressures, at the same depth, are equal, the downward pressures in the two tubes are also equal. 

Con. 1. Hence in different vessels, containing d. fferent fluids, the pressures are as the areas of the 
bases multiplied into the depths, and specific gravities. 

Con. 2. If a cone be filled with a fluid, and standing on its base, the pressure on its base will be equal 
to three times the weight of the fluid. Let 13 be equal to the base, H equal to the perpendicular height, 


then the solid content, or weight, will be equal ?-x H, but the pressure will be B x II, therefore equal 

J 


to three times its weight. 

Cor. 3. A small quantity of fluid may be made to press with a force sufficient to raise a great weight. 

Since (as was shown in Prop. V.) as much fluid as will fill the tube DBIV presses upward against 
VM, with a force equal to the weight of as much fluid as would fill the space BHVM ; the base remain¬ 
ing the same, the space BHVM, that is, the weight which may be raised, will (by this Prop.) be as the 
height VB, which may be increased at pleasure. 

Exp. Let two circular pieces of wmod be united by leather in the manner of a pair of bellows ; in 
the upper board insert a long tube with a large bore; through which pour water into the vessel; the 
upward pressure of the water, as it is poured in, w ill raise a great weight. 

Cor. 4. From hence it may be proved, independently of the reasoning in Prop. V. that, in bended 
vessels, or channels of any form, fluids rise to the same height, whatever be the difference between the 
quantities of fluid on each side ; for whatever be the form of the channels, the plane which is perpen¬ 
dicular to the lowest point being considered as the common base, the pressure upon it is equal, when 
the fluid on each side is of equal altitude ; and the whole mass can only be at rest when the opposite 
pressures are equal. 

Schol. This pressure of the fluid upon the base does not alter the weight of the vessel and fluid con¬ 
sidered as one mass, because the action and re-action which cause it, with respect to the weight of the 
vessel, destroy each other; the vessel being as much sustained by the action upward, as it is pressed 
by the re-action downward. 


PROP. \ II. The pressure of a fluid upon any indefinitely small part of the side of 
a vessel which contains it, is equal to the weight of a column of the same fluid, whose 
base is the part pressed, and whose height is the distance of that part from the surface 
of the fluid. 

Let ABCD be a vessel filled with fluid; AB its surface; and L a point in the side of the vessel. 
The indefinitely small drop w r hich lies next to the point L is pressed downward (by Prop. I.) by a force 
equal to the weight of a column of water whose base is L, and height LA, the distance of that part from 
the surface. And (by Prop. III.) this di’op is pressed sideways toward L with the same force with which 
it is pressed downward. Whence the position is manifest concerning the point L. And the same may 
be proved concerning any other points M, N, C, equal to L. The same is evidently true in an inclined 
vessel. 


Plate 5. 
Fig. 12. 


PROP. VIII. The pressure of a fluid upon any plane is equal to the weight of a 
body which has the same density with the fluid, and is formed by raising perpendicu¬ 
lars upon each indefinitely small part of the plane, equal in height to the distance of 
that part from the surface of the fluid. 

It has been proved, in the last proposition, that the pressure upon each indefinitely small part of the 
line AC, in the side of the vessel ABCD, is equal to the weight of a column of fluid whose base is the 
part pressed, and whose height is the distance of that part from the surface AB. Hence if from the 
point L a perpendicular LO be raised whose base is L, and whose length LO is equal to LA’the distance 
of L from the surface, if this perpendicular consisted of matter of the same density with the fluid in the 
vessel, the weight of this perpendicular column would be equal to the pressure upon the point L If in 
like manner, perpendiculars, consisting of matter of the same density with the fluid, were raided upon every 
point between A C, they would together fill up the area of the triangle ACD ; and the pressure upon 
the whole line AC in the side of the vessel ABCD, because it is equal to the sum of the pressures upon 


> 


Ciiap. I. 


THE PRESSURE OF FLUIDS. 


53 


all its parts, must be equal to the weight of this triangle ACD. The same may be proved concerning 
any other lines in the side of the vessel, as HI, EF. Consequently, the pressures upon the whole side , 
will be equal to the we.ght of as many such triangles as there can be lines drawn upon it in the same * • 
manner as AC, HI, EF, are drawn. But all these triangles together would till up the whole space, or 
compose a solid, CFGDAE. Therefore the pressure upon the side AECF will be equal to the weight 
of this solid, consisting of matter which has the same density with the fluid in the vessel; which solid 
is formed by raising perpendiculars upon each line of the side, respectively equal to the distance of 
that point from the surface of the fluid. 

in like manner, if AC is a line drawn in the inclined side of a vessel, in which the water reaches Plate 5. 
to the level AB, the pressure upon this line may be estimated as before. SL is the distance of L from Fig. 13 
the surface. Let therefore a perpendicular LO, equal in length to LS, be raised upon the point L ; 
then, if this perpendicular were a column of matter of the same density with water, the weight of it 
would be equal to the pressure upon L. For the same reason, if a perpendicular MP is raised upon the 
point M, and is made equal in length to MT, the distance of M from the surface ; such a perpendicular, 
consisting of matter of the same density with water, and being of the same size would have the same 
weight as the column of water MT. And since (by Prop. I.) the pressure upon M equals the weight of 
the incumbent water MT, it likewise equals the weight of the perpendicular MP. In like manner, the 
points N and C are pressed by the weight of the incumbent columns NV and XC, which is equal to the 
weight of the perpendiculars NQ,, CR, supposing those perpendiculars to be equal in height to NV and 
XC, and to consist of matter whose density is the same with that of the columns NV and XC. Thus the 
pressure upon the whole line, being made up of the pressures upon all its parts, will be equal to the 
weight of as many perpendiculars, as can be raised in this manner between A and C. The sum of all 
those perpendiculars is the triangle ACR, whose weight therefore is equal to the pressure upon the 
line AC. But if as many such triangles were added together, as there are lines parallel to AC in the 
whole side of the vessel, all these triangles together would form a solid. And since this solid is the 
sum of all the pressures upon each point of the side, the weight of it, supposing it to consist of matter 
that has the same density as water, would be equal to the pressure upon the whole side. 

PROP. IX. The pressure upon any one side of a cubical vessel; filled with fluid, is 
half the pressure upon the bottom. 

The bottom sustains a pressure equal to the whole weight of the fluid in the vessel. And the press- plate. 5 
ure which the side sustains is equal to the weight of the prism CFGDAE, which (El. XI. 28.) is half the Fig 14. 
cube ; therefore the side sustains a pressure equal to half the pressure upon the bottom. 

Or thus; Because the pressure upon every part of the vessel at the bottom is equal to the weight of 
a column whose base is the part pressed upon, and height that of a perpendicular from the bottom to 
the surface; if the pressure were the same every where from the top to the bottom, it would be equal 
to the weight of as many such columns as would correspond to all the parts of the vessel. But the pressure 
every where diminishes as we approach toward the surface, where it is nothing; the pressure on the 
side is therefore only half of that on the bottom of the vessel; a number of terms in arithmetical pro¬ 
gression beginning from nothing being half the sum of an equal number of terms, each of which is equal 
to the last in the progression. 

Cor. 1 . The gravity of the fluid in a cubical vessel producing upon each of the four sides a pressure 
equal to half that upon the bottom, and upon the bottom a pressure equal to itself, produces on the whole 
a pressure three times as great as itself. 

Cor. 2. When the area of the part pressed is given, the pressure is as the perpendicular distance of 
that part from the surface ; where the depth of the part is given, the pressure is as the area. 

Schol. There is a particular point in which the whole pressure against the side acts; it is called 
the centre of pressure , and is the same with the centre of oscillation of the side vibrating on the upper 
line of it as an axis. See Prop XLVI. Schol. 1. Book II. 


54 


OF HYDROSTATICS. 


Book III. 


Plate 5. 
Fig. 10. 


CHAPTER II. 

Of the Motion of Fluids. 

SECTION I. 

Of Fluids passing through the Bottom or Side of a Vessel. 

PROP. X. The momentum with which any fluid runs out of a given orifice in the 
bottom or side of a vessel, is proportional to the perpendicular depth of the orifice 
below the surface of the fluid. 

The pressure of a fluid against any given surface being (by Prop. I. and III.) proportional to the 
perpendicular height of the fluid above that part; if that given surface be removed, the fluid will be 
driven through the orifice by this pressure. The force therefore with which the fluid passes through 
the orifice is as the perpendicular depth of the orifice below the surface of the fluid ; but the momen¬ 
tum is always as the moving force ; therefore the momentum is also as the perpendicular depth of the 
orifice. 

PROP. XI. The momentum with which any fluid runs out of a given orifice in 
the bottom or side of a vessel, is as the square of its velocity, or as the square of the 
quantity of matter. 

The momentum (by Book II. Prop. XI.) is in the compound ratio of the quantity of matter and ve¬ 
locity. And it is manifest, that, since the orifice is given, the quantity of fluid discharged will 
always be as the velocity; therefore the momentum is as the square of the velocity, or of the quantity 
of fluid. 

PROP. XII. The velocity with which any fluid runs out of an orifice in the bottom 
or side of a vessel, is as the square root of the perpendicular depth of the orifice from 
the surface of the fluid. 

Because the momentum is as the square of the velocity, (by Prop. XI.) and as the perpendicular 
depth of the orifice (by Prop. X.) the square of the velocities (El. V. 11.) is as the perpendicular depth, 
and, consequently, the velocity as the square root of the perpendicular depth. 

Cor. 1 . Hence a fluid running out of a vessel which empties itself, and whose horizontal sections 
are all equal, flows with an uniformly retarded velocity ; for the perpendicular depths are continually 
diminishing. 

Cor. 2. Hence also the surface descends with an uniformly retarded velocity, and the spaces describ¬ 
ed by it, in equal portions of time, are (Prop. XXVIII. Book II.) as the odd numbers 1, 3, 5, 7, 9, & c . 
taken backward. 

Cor. 3. If therefore a cylindrical vessel be divided into portions, continued to the surface of the 
fluid, which are as the odd numbers, 1, 3, 5, 7, &c. a clepsydra or hour-glass will be formed ; for the 
surface will descend through these divisions in equal times. 

PROP. XIII. A fluid runs out of an orifice in the bottom or side of a vessel, with 
the velocity which a heavy body would acquire in falling freely through a space equal 
to the perpendicular distance of the orifice from the surface of the fluid. 

Let ABCD be a vessel filled with any fluid, to the height FG. It is manifest, that at the beo-innin®- 
of the fall of each drop from the upper surface FG, it must be carried downward by its gravity with 
the same velocity with which any other heavy body would begin to descend. And, if an orifice be 
made in the vessel at L, any point below the surface, the fluid which passes through that orifice will 
(by Prop. XII.) move with a velocity which is as the square root of the distance from the surface. But 
if a body were to fall from the surface to the point L, it would acquire a velocity which would be (by 
Book II. Prop. XXVI. Cor. 2.) as the square root of this distance. Therefore, since the velocity with 
which the fluid moves is, at the beginning of its motion, equal to that of a falling body, and since at 
every given distance these velocities have the same ratio, namely, that of the square root of the dis¬ 
tance from the surface, that is, (El. V. 9.) are equal, the proposition is manifest. 


Chap. II. 


OF THE MOTION OF FLUIDS. 


55 


Cor. Supposing O, V, T, Q, to represent the area of the orifice, velocity, time, and quantity flow¬ 
ing out in that time, respectively ; Q, will vary as O x V x T, or as O x T X >/R? (Prop- XII.) and 
when T is given, as O x v'R- . 

Schol. When a fluid spouts from a vessel, it rushes from all sides toward the orifice, which is the 
cause of the contraction of the stream at the distance from the orifice equal to its diameter, and is call¬ 
ed the vena contracta. Now the area of the orifice is to the area of the smallest section of the stream, 
nearly as v/2 to 1 ; hence (by Prop. IV.) the velocity at the vena contracta is to the velocity at the 
orifice as v/2 to 1. Sir I. Newton found, that the velocity at the vena contracta, was that which a body 
acquires in laliing down the altitude of the fluid above the orifice. We must, therefore, distinguish be¬ 
tween the velocity at the orifice, and at the vena contracta, and in the doctrine of spouting fluids, it is 
the latter velocity which must be considered, and the point of projection must be assumed lrom 
that point. 

PROP. XIV. When two cylindrical vessels have their bases, heights, and orifices 
equal, if one of them be always kept full, it will discharge double the quantity of fluid 
discharged in the same time by the other whilst it empties itself. 

For (by Prop. I.) the fluid will continue through the whole time, to run with the same velocity out 
of the vessel that is kept full. But the fluid will run (Cor 1. Prop. XII.) with an uniformly retarded 
velocity out of the vessel which empties itself. And, since both vessels are full at first, the velocity 
which continues uniform in one vessel, will (by Prop. I.) be the same with the first velocity in the 
vessel in which the fluid is uniformly retarded. Therefore the quantity discharged out of the former 
vessel will be to the quantity discharged in an equal time out of the latter, as the space described by a 
body moving with an uniform velocity, to the space described by a body which sets out with the same 
velocity, and is uniformly retarded. But (by Book I. Prop. XXVII.) the space described by the former 
will be double of the space described by the latter. Therefore the quantity discharged out of the 
former vessel, will be double of the quantity discharged out of the latter. 

PROP. XV. A stream of any fluid which spouts obliquely forms a parabola. 

Each drop in a stream of fluid, spouting obliquely, is a heavy body projected obliquely by the force 
or pressure which drives it out of the orifice. Therefore (by Book II. Prop. LVIII.) every drop of the 
stream, that is, the whole stream forms a parabola. 

Exp. Observe the figure formed by a fluid spouting obliquely. 

Cor. Hence fluids spouting obliquely are subject to the laws of projectiles laid down, Book II. Ch. 
VII. Sect. 1. 

PROP. XVI. When a fluid spouts horizontally from an orifice in the side of a 
vessel which is kept full, if a line passing through the orifice perpendicular to the 
horizon, and intercepted between the surface of the fluid and the horizontal plane that 
receives it, be made the diameter of a circle, and a line drawm horizontally from the 
orifice to the circumference, the distance, to which the fluid will spout, will be double 
of this horizontal line. 

Let AB be the perpendicular; C, E, or e, the orifice; ADHB the semicircle drawn on the side; 
ED, CH, d e, lines drawn horizontally from the orifice to the circumference. The fluid spouts at E 
(by Prop. XIII.) with the velocity which a heavy body would acquire in falling from A to E ; and this 
motion, being in a horizontal direction, can neither be accelerated nor retarded by the force of gravita¬ 
tion, and will therefore continue uniform. But beside this, the fluid spouts with the velocity which it 
acquires in falling after it has passed the orifice. This velocity, when the fluid arrives at GB, is the 
same with that which any other heavy body would have acquired in falling through an equal space from 
E to B. Let this velocity be called the descending velocity, and that with which the fluid spouts at E 
the horizontal velocity. Then, since the horizontal velocity is the same with that which a body would 
acquire by falling from A to E, and the descending velocity, when the fluid arrives at the plane GB, is 
the same with that which a body would acquire by falling from E to B, and since (by Book II. Prop. 
XXVI.) the spaces AE, EB, described by falling bodies, are as the squares of the last acquired velocities 
of bodies falling through them; that is, (inverting the terms) the squares of these last acquired veloci¬ 
ties, or the squares of the horizontal and descending velocities, are as the lines AE, EB. But in the 
triangle ADB, right-angled (El. VI. 8.) at D, DE is a mean proportional between AE, EB, and the 
square of ED is to the square of EB, as AE is to EB. But the square of the horizontal velocity is to 
the square of the last descending velocity, as AE to EB. Therefore the square of the horizontal velocity 
is to the square of the last descending velocity as the square ED to the square EB; whence the horizontal 


Plate 5 
Fig. 11 


56 


OF HYDROSTATICS. 


Book. III. 


Plate 5. 
Fig. 11. 


Plate 5. 
Fig. 15. 


velocity is to the last descending velocity as ED to EB. Now the spaces described in the same time, 
in uniform motions, are ^Book II. PROF. VI.) as the velocities. Consequently, if tne fluid had begun 
to fall from E with the velocity it has acquired at B, and had fallen uniformly, in the time of descent 
the spaces described by the horizontal and descending velocities would have been respectively as those 
velocities; that is, as ED to EB. Thus while the fluid was descending till it reached the plane GB, 
the horizontal velocity would have carried it forward through a space equal to ED, or the horizontal 
distance would be ED. But the descending velocity being at the first nothing, and continually increas¬ 
ing, the time of descent (see Book II. Prop. XXVII.) is twice what it would have been upon the suppo¬ 
sition that it began to descend from the last acquired velocity. And the horizontal velocity is uniform, 
and therefore in twice the time, or the true time of descent, the fluid will be carried horizontally to 
twice the distance ED. Consequently, if BF be made equal to twice DE, whilst the stream is descend¬ 
ing from E to GB, it will be carried forward to the point F. The same may be proved concerning any 
other points, C, e. 

PROP. XVII. If a fluid spout horizontally out of orifices in the side of a vessel 
which is kept full, it will spout to the greatest distance from the orifice which is in the 
middle of the side, and to equal distances from orifices equally distant from the middle. 

Let C be the orifice in the middle of the side, and E, c, equal orifices at equal distances from C. 

The distance to which the fluid will spout at C (by Prop. XVI.) is twice CH, and at E twice ED. 
But CH (El. III. 15.) is greater than DE, any line drawn from the diameter parallel to the radius; 
therefore twice CH is greater than twice ED. 

Also since the horizontal distances to which the fluid will spout at E and e, are twice ED, or erf; 
and that ED, erf, being equally distant from the centre, and parallel to the radius, (El. III. 14.) are 
equal; the horizontal distances from E, e, are equal. 

Hence, if in the plane of the horizon, GB be drawn perpendicular to the side AB, and GB be 
double of CH, and FB double of DE, or rfe, the fluid spouting from C will fall upon G, and from E and 
e, upon F. 

Cor. If the side of the jet d’eau be inclined, in any angle to the horizon, and the direction, and ve¬ 
locity of the spouting fluid be known, the amplitude, altitude, and time of flight, may be discovered by 
the rules investigated in Book II. on Projectiles. 

Exp. Let water spout from the middle orifice, and from orifices equally distant from the middle, the 
truth of the proposition will be manifest. 

Remark. In all propositions respecting the times in which vessels empty themselves, the orifice is 
supposed to be very small in respect to the bottom of the vessel, otherwise the experiments do not 
agree with the theory. 

Def. II. A river is a stream of water which runs by its own weight down the inclin¬ 
ed bottom of an open chanel. 

Def. III. A section of a river is an imaginary plane, cutting the stream, which is 
perpendicular to the bottom. 

Def. IV. A river is said to flow uniformly when it runs in such a manner, that the 
depth of the water in any one part continues always the same. 

PROP. XVIII. If a river flows uniformly, the same quantity of water passes in an 
equal time through every section. 

Let AB be the reservoir, BC the bottom of the river, and ZX, QR, sections of the river. Because 
the river flows uniformly, the same quantity of water which passes through ZX in a given time must 
pass through QR in the same time ; otherwise the quantity of water in the space ZQXR, must in that 
time be increased or diminished, and consequently the depth of the water in that space altered; con¬ 
trary to the supposition. 

Cor. Hence if V, B, D, be the velocity, breadth, and depth respectively, V x B x D will be a 
given quantity, and V will vary as — — 

PROP. XIX. The breadth of the channel being given, the water in rivers is accele¬ 
rated in the same manner with any body moving down an inclined plane. 

For each drop of the water moves down upon the inclined plane of the bottom, or upon the inclined 
plane of the sheet of water, next below it, parallel to the bottom. 


* 



CHAP. II. 


OF THE MOTION OF FLUIDS. 


PROP. XX. The breadth of the channel being given, the velocity of each drop of 
water in a river is the same that a body would acquire in falling from the level of the 
surface of the water in the reservoir, to the place of the drop. 

Let AB be the depth of the reservoir, AP the level of its surface, and BC the bottom of the channel. 
Any drop at E, after it comes out of the reservoir at K (by Prop. XIX.) rolls down the inclined plane 
KE, parallel to the bottom. And this drop, when it comes out of the reservoir AB at K (by Prop. XIII.) 
has the same velocity which a heavy body would acquire in falling from A to K; and, in rolling down 
the inclined plane KE, it acquires (by Book II. Prop. XXXIV.) the same velocity which any heavy 
body would acquire in falling down GE, the perpendicular height of the plane. At E the drop will 
therefore have acquired a velocity equal to that which a body would acquire by falling through AK 
and GE, that is, through MGE, the perpendicular drawn from the level of the reservoir to the place of 
the drop. 

Cor. 1 . Hence the breadth of the channel being given, the velocity of each drop of water in a river 
is as the square root of its distance from the level of the surface of the reservoir. For, if E and Ii be 
two drops in different parts of the river, and AP the level, the velocity of the drop E is the same that 
a body would acquire by falling down ME, and that of R the same which a body would acquire by falling 
down HR. Therefore (by book II. Prop. XXVI. Cor. 2.) the velocity of the drop E is to the velocity 
of R, as the square root of ME to the square root of HR. 

Cor. 2. Hence the breadth of the channel being given, the water at the bottom of a river will run 
faster than the water at the surface. 

PROP. XXI. The breadth of the channel being given, the depth of the river con¬ 
tinually decreases as it runs. 

The same quantity of water (by Prop. XVIII.) passes through each of the sections ZX, QR, in the 
same time. But (by Prop. XX. Cor. 2.) the water runs faster at the lower section QR, than at the 
upper ZX. Therefore the area of the section QR must be as much less than the area of the section 
ZX, as the velocity at QR is greater than the velocity at ZX. But the breadth of the sections are by 
supposition equal; therefore their areas are (El. VI. 1.) as their heights. Consequently the heights of 
the sections QR, ZX, wall be inversely as the velocities at those sections ; that is, the depth of the 
water at QR will be as much less than the depth at ZX, as the velocity at QR is greater than the 
velocity at ZX. 

PROP. XXII. At a given distance from the reservoir, if the river flow uniformly, 
the velocity of the water will be inversely as the breadth of the channel. 

Because the river flows uniformly, the depth at any given section ZX is always the same ; and in 
any given time, the same quantity of water must flow through the different sections ZX, QR, as was 
shown in Prop. XVIII. But a given quantity of water cannot flow in a given time through any section, 
unless as much as the area is increased, so much the velocity is diminished, and the reverse ; that is, 
the velocity must be inversely as the area of the section, or the depths being given, as its breadth.* 

PROP. XXIII. The depth of a river being given, the pressure upon any part of 
the bank will be the same, whatever is the breadth of the river. 

The pressure upon any given part in the bank (by Prop. I. and III.) will be as the distance of that 
part from the surface; which remains the same whilst the depth is the same, whatever be the breadth 
of the river; therefore the pressure will remain the same. 

PROP. XXIV. If the breadth of a river be given, the pressure on any part of the 
bank will be as the depth of the river. 

For the pressure on any part of the bank is (by Prop. I. and III.) as the depth of that part below the 
surface, which depth will increase with the depth of the river. 

PROP. XXV. The pressure against any given surface in the bank of a river, if 
that surface reaches from the bottom to the top of the stream, is equal to the weight of 
a column of water whose base is the surface, and whose height is half the depth of 
the stream. 

* This and the three preceding propositions can be applied only to straight regular canals of considerable declivity and 
no great length. 


58 


OF HYDROSTATICS. 


Book III. 


Plate 5. 
Fig. 15. 


Flate 5. 
Fig. 11. 


Let ZQXR be a given surface in the bank, reaching from the bottom BC ol the river to its top AD. 
The pressure upon this is (from what was shown in Prop. IX.) hail the pressure on an equal surface at 
the bottom XR; which pressure (by Prop. I. and Ill.) is equal to the weight of a column ol water 
whose base is the surface ZQ, and whose height is the depth of the stream. Therefore the pressure 
against the surface ZQXR is equal to the weight of a column whose base is the surface ZQ, and its 
height half the depth of the stream. 

PROP. XXVI. When a stream which moves with the same velocity in every part 
strikes perpendicularly upon any obstacle, the force witli which it strikes is equal to 
the weight of a column of the same fluid, whose base is the obstacle, and whose height 
is the space through which a body must fall to acquire the velocity of the stream. 

Let a stream of water How horizontally out of the oritice e. If this stream were to strike upon an 
obstacle of the same breadth every way as the oritice or stream, placed perpendicular to the horizon, 
the stream must strike upon the obstacle with its whole force. But this force is equal to the weight 
of a column of water whose base is e, and height A e . And (by Prop. XIII.) A e is the height from 
which a body must tall to acquire the velocity with which the stream spouts from e. Therefore the 
force with which this stream would strike such an obstacle is equal to the weight of a column ol water 
whose base is e, and height that from which a body must fall to acquire the velocity of the stream. 
And because no part of the stream, however broad, can strike the obstacle except so much as is contain¬ 
ed within a section equal to the surface of the obstacle, no other part of the stream is to be considered 
in estimating this force. It is also manifest, that if the stream flow horizontally with the same velocity, 
in any other manner than through an orifice, as in the current of a stream, it will strike an obstacle 
with the same force. 

PROP. XXVII. When the obstacle is given, the force with which a stream strikes 
upon it will be as the square of the velocity with which the stream moves. 

If any stream strike upon a given obstacle, the force will (by Prop. XXVI.) be equal to the weight 
of a column of water whose base is the obstacle, and whose height is equal to the space through which 
a body must fall to acquire the velocity of the stream. Since then the base is given, the weight will be 
as the height of such a column. But the spaces through w'hich bodies fall to acquire different velocities 
are (by Book II. Prop. XXVI.) as the squares of those velocities. Therefore the height of this column, 
and its weight, and consequently the force of the stream, which is equal to this weight, will be as the 
square of the velocity with which the stream moves. 


CHAPTER III. 

Of the Resistance of Fluids. 

PROP. XXVIII. If a spherical body is moving in a given fluid, the resistance which 
arises from the reaction of the particles of the fluid is, within certain limits of the ve¬ 
locity, as the square of the velocity with which the body moves. 

A spherical body moving in a given fluid, the number of particles which it will meet within a given 
time will be as its velocity; for the space through which it will pass will be as its velocity, and the 
number of particles it will meet with w ill be as the space through which it passes. But the reaction 
of the particles of the fluid, and consequently the resistance, is as the number of particles or quantity of 
matter by which the resistance is made. Again, if a given quantity of matter is to be moved, the mov¬ 
ing force is*(by Book II. Prop. IX.) as the velocity communicated ; and the resistance of that given 
quantity of matter is as the moving force. Therefore the resistance arising from reaction in a given 
number of particles of fluid is as the respective velocities with which they are moved ; that is, as the 
velocities with which the bodies which pass through the fluid move. The resistance of the fluid 
being then as the velocity on a double account, first, because the number of particles moved are as the 
velocity of the moving body, and secondly, because the resistance of a given number of particles is 
as the velocity of the moving body ; the resistance will be in the duplicate ratio, or as the square of 
this velocity. 

Schol. In very swift motions, the resistance of the air increases in a greater ratio ; (see Remark to 
Prop. LV III. Book II.) and in other fluids the same consequence w'ould follow’ for the same reason,- with 



Chap. III. 


THE RESISTANCE OF FLUIDS. 


59 


respect to projected bodies. Besides, the greater the velocity is, the less will be the pressure against 
the back of the body which will cause a deviation in the law of resistance. 

PROP. XXIX. When a spherical body moves with a given velocity in any fluid, 
the resistance of the fluid arising from its reaction will be as the square of the diameter 
of the spherical body. 

A spherical body, in moving through a fluid, displaces a cylindrical column of that fluid, the height 
of which is the space which the sphere describes, and its base a great circle of the spherical body. 
Because the velocity is given, the space described in a given time, that is, the length of the column is 
given; whence, the quantity of fluid in the column, that is, the. column will be as its base, a great cir¬ 
cle of the sphere. And the resistance which the column of fluid makes by reaction to the motion of the 
sphere will be as its quantity of matter; it will therefore be as the base of the column, or as the great 
circle of the sphere, or (El. XII. 2.) as the square of its diameter. 

PROP. XXX. If two unequal homogeneous spheres are moving in the same fluid 
with equal velocities, the greater sphere will be less resisted in proportion to its weight, 
than the lesser sphere. 

The weights of spheres, or their solid contents, are (El. XII. 18.) as the cubes of their diameters; 
but their resistances (Prop. XXIX.) are as the squares of their diameters; and the cubes of any numbers 
have a greater ratio to each other than their squares. Therefore the ratio of the weights of spherical 
bodies is greater than that of their resistances in a given fluid; that is, the weight of the greater sphere 
exceeds the weight of the lesser, more than the resistance of a given fluid against the former exceeds 
the resistance against the latter, provided the spheres are moving with equal velocities. 

Schol. Hence the resistance of the air may be able to support small particles of fluid, but unable to 
support them when they are collected into larger drops. 

PROP. XXXI. The resistance of a fluid, arising from its reaction, is as the side of 
a body perpendicularly opposed to it. 

The resistance is as the column, or quantity of fluid removed in a given time, which, as was shown, 
(Prop. XXIX.) is as the base of the column ; that is, as the side of the body perpendicularly opposed 
to it. 

PROP. XXXII. When equal spheres move with the same velocity in different 
fluids, the resistances will be as the densities of the fluids. 

The resistances arising from reaction are as the momenta communicated to the fluid in a given 
time ; that is, since the spheres move with equal velocities, as the quantities of matter moved. But be¬ 
cause the spheres are equal, the bases of the columns to which they communicate motion are equal; and 
because the spheres move with equal velocity, the lengths of the columns to which they communicate 
motion, are equal. Hence the columns to which motion is communicated, having their bases and heights 
equal, are of equal magnitude ; and consequently, their quantities of matter are as their densities. But 
it has been shown, that their momenta and resistances are as their quantities of matter; therefore their 
resistances are as their densities. 

Schol. Hence drops of water may be sustained in the lower parts of the atmosphere, which cannot 
be sustained in the higher. 

PROP. XXXIII. The retardation of bodies in a resisting fluid, where the weights 
of the bodies are given, is as the resistance of the fluid. 

The more a body is resisted by any fluid in which it moves, the greater portion of its momentum 
is destroyed; but, because the weight of the body is given, its momentum is as its velocity; therefore 
the greater the resistance of the fluid, the greater portion of its velocity is destroyed, that is, the more 
it is retarded. 

PROP. XXXIV. When the resistance is given, the retardation is inversely as the 
weights. 

The same resistance will destroy an equal portion of momentum whatever is the weight of the mov¬ 
ing body. But when the momentum is the same, the velocity is (by Book II. Prop. XII.) inversely as 


60 


OF HYDROSTATICS. 


Book III. 


the quantity of matter. Therefore the velocity destroyed, or the retardation, will be inversely as the 
quantity of matter in the body in which the momentum is destroyed; and the weight is as the quantity 
of matter; therefore the retardation is inversely as the weight. 

PROP. XXXV. The retardation of spherical bodies, moving with equal velocities 
in the same fluid, is inversely as their diameters. 

The resistance which spherical bodies meet with in a given fluid is (by Prop. XXIX.) as the squares 
of their diameters. The retardation, when the weight is given, is (by Prop. XXXlII.)as the resistance; 
and when the resistance is given, the retardation (by Prop. XXXV.) is inversely as the weight; that is, 
(El. XII. 18.) inversely as the cubes of the diameters. Now, when unequal spheres move with the 
same velocity in the same fluid, the retardations will be unequal, both because the resistances are un¬ 
equal, and because the weights are unequal. The retardations will therefore be directly as the squares 
of the diameters, and inversely as the cubes of the diameters; that is, (compounding these ratios) in¬ 
versely as the diameters. 

PROP. XXXVI. When a body moves in an imperfect fluid which has tenacity, or 
the parts of which cohere, the resistance of any given portion of the fluid from this 
cause, is inversely as the velocity of the body ; the resistance, when the velocity is 
given, is as the quantity of fluid through which the body passes ; and the resistance 
is always as the time during which the body moves in the fluid. 

Case 1. Suppose such an imperfect fluid, as soft clay, divided into thin plates; each plate having a 
certain portion of tenacity will continue to resist the body during the whole time in which it is passing 
through it; the resistance therefore will be less, the shorter time the body takes in passing through it, 
that is, the greater velocity the body moves with. And this is true concerning every plate which com¬ 
poses the fluid. Therefore the resistance arising from tenacity in a given quantity of fluid, is inversely 
as the velocity of the body which passes through it. 

Case 2. Again, the velocity of the body being given, the resistance which the body meets with from 
what has been said, is also given, and will be as the number of plates or quantity of the fluid. 

Case 3. Lastly, when a body moves for a given time, the resistance (by the second case) is as the 
number of plates, that is, as the space through which it passes in a given time, that is, (by book II. 
Prop. VI.) as the velocity directly. And (by the first case) the resistance is, on account of the 
tenacity, inversely as the velocity. Therefore as much as the resistance is increased on account of the 
velocity in one respect, so much it is diminished on account of the velocity in another; and conse¬ 
quently, whatever be the velocity of a body in such a fluid, the resistance which it meets with in a 
given time will be the same ; whence this resistance will be as the time in which the body moves 
in the fluid. 


CHAPTER IV. 

Of the Specific Gravities of Bodies. 

Def. V. The density of a body is its quantity of matter when the bulk is given. 

Def. VI. The specific gravity of a body is its weight, compared with that of another 
body of the same magnitude. 

Cor. 1 . The specific gravity of a body is as its density. For the specific gravity of a body is the 
weight of a given magnitude, and the weight of a body (by Book 11. Prop. XXIV. Cor.) is as its quan¬ 
tity of matter; therefore the specific gravity of a body is as the quantity of matter contained in a «iven 
magnitude, that is, as its density. 

Cor. 2 . The specific gravities of bodies are inversely as their magnitudes when their weights are 
equal. For by the last Cor. the specific gravities of bodies are as their densities, and their densities 
(from Def. 1.) are inversely as their magnitudes when their weights are equal. Therefore the specific 
gravities are also inversely as their magnitudes when their weights are equal. 

PROP. A. The weight of a body varies as its magnitude and specific Gravity 
conjointly. 



Chap. IV. 


OF SPECIFIC GRAVITIES. 


61 


For if the magnitude of any body is varied, its specific gravity remaining the same, the weight must 
be altered in the same ratio. And if the specific gravity vary while its magnitude continues the same, 
the weight must also vary in the same ratio. Therefore the weight must vary as the magnitude and 
specific gravity conjointly. 

PROP. XXXVII. A fluid specifically lighter than another fluid will float upon its 
surface. 

For (by Book II. Prop. XXIV.) the lighter fluid will be less powerfully acted upon by the force of 

gravitation than the heavier; whence, the heavier will take the lower place. 

Exp. 1 . Let a small and open vessel of wine be placed within a large vessel of water ; the wine will 
ascend. 

2. Let mercury, water, wine, oil, and spirits of wine, be put into a phial in the order of their spe¬ 
cific gravities ; they will remain separate. 

PROP. XXXVIII. The heights to which fluids, which press freely upon each other, 
will rise, are inversely as their specific gravities. 

Since (by Prop. VI.) the opposite parts of a homogeneous mass of fluid, in a curved tube or channel, 
press equally against each other when they rise to the same height; in order to preserve the pressure 

equal when the fluids on each side are different, that which has the least specific gravity must propor¬ 

tionally rise above the level to preserve the balance; and the reverse. 

Exp. Into the longer arm of a recurved tube, of equal bore throughout, and open at each end, pour 
such a quantity of mercury, that it shall rise in each arm about half an inch ; then pour water into the 
longer arm till the mercury is raised one inch above its former height; the specific gravities of these 
fluids will be inversely as the heights to which they rise. 

PROP. XXXIX. The force with which a body lighter than any fluid endeavours to 
ascend in that fluid, is as the excess of the specific gravity of the fluid above the solid. 

Since ABCD, the fluid in a vessel, will be at rest (Prop. III.) when every part of an imaginary 
plane SQ, under the surface of the floating body p t e i, sustains an equal pressure; if the solid body be 
of equal specific gravity with the fluid, that is, weighs as much as a quant.' y of the fluid equal to it in 
bulk, and whose place it takes up, this imaginary plane being equally pressed by the solid, as if the 
same space were filled with fluid, the fluid will be at rest, and the solid will neither ascend nor descend. 
Consequently, if the body be specifically heavier than the fluid, that part of the plane which is directly 
under the solid being so much more pressed than the other equal parts of the same plane as the solid 
body is specifically heavier than the fluid, the body must descend with a force equal to that excess; 
and, on the contrary, if the body be specifically lighter than the fluid, that part of the plane which is 
directly under the solid being so much less pressed than the other equal parts of the same plane, as the 
body is specifically lighter than the fluid, it must be buoyed up with a force equivalent to the difference 
of their specific gravities. 

PROP. XL. Any fluid presses equally against the opposite sides of a solid body im¬ 
mersed in it. 

The opposite sides of the solid are at the same depth; and fluids at the same depth press equally. 
Thus the opposite sides RM, SN, of any body immersed in a vessel of water ABCD, are pressed equally 
by the surrounding fluid. 

Cor. No motion of the solid will be produced by these opposite lateral pressures. 

PROP. XLI. A body immersed in a fluid is pressed more upward than it is down¬ 
ward, and the difference of these two pressures is equal to the weight of as much of the 
fluid as would fill the space which the body fills. 

The body MRNS being immersed in a vessel of water ABCD, its lower part MN must be pressed 
upward just as much as the water itself, at the same depth MNT, would be if no solid were immersed. 
Now the water at any depth (by Prop. III.) is pressed as much upward as it is pressed downward. And 
at the depth MNT, the portion of this stratum MN would, if the solid were away, be pressed down¬ 
ward by a force equal to the weight of the incumbent column of wat^r EMNH. Therefore the force 
with which MN, that is, the lower part of the solid, is pressed upward, is equal lo the weight of as 
much water as would fill the whole space EHMN. But the solid body RSMN is pressed downward by 
the weight of the column above it EHRS. Therefore the difference between the two pressures is the 


Plate 5. 
Fig. 19 


Plate 1. 
Fig. IS. 


Plate {j. 
Fig. 18. 


OF HYDROSTATICS. 


Book III. 


difference of the weights of the two columns of water EHMN, and EHRS; that is, the upw T ard pressure 
upon the solid body RSMN exceeds the downward pressure, by a force equal to the weight ol as much 
water as would fill the space RSMN, taken up by the solid body. The case will be the same, whatever 
is the figure of the body immersed. 

PROP. XLII. A body immersed in a fluid, if it be specifically heavier than the flu¬ 
id, will sink. 

If the body RSMN is specifically heavier than the fluid, it w'eighs more than a quantity of the fluid 
of the same bulk with it. Hence the column EHMN, consisting of the column of fluid EHRS and the 
solid body RSMN, is heavier than the same column would be if it consisted wholly of w'ater. But the 
upward pressure against MN is (by Prop. III.) equal to the downward pressure ot the column of water 
EHMN, and therefore only sufficient to support the weight of that column. It cannot then support the 
weight of the heavier column, consisting of a fluid and a solid, EHMN; and that part ot this column 
which is specifically heavier than the fluid, that is, the solid will sink, with a force equal to the differ¬ 
ence of the weights of the column of fluid EHMN, and the mixed column EHRS, RSMN. 

PROP. XLIII. A body specifically lighter than the fluid, in which it is immersed, 
will rise to the surface and swim. 

If the solid RSMN be a body specifically lighter than water, the column EHMN will weigh less as 
it consists of the column of water EHRS, and the solid RSMN, than if it consisted entirely of water. 
Consequently, the upward pressure upon MN, which is equal to the weight of the column of water 
EHMN, will be equal to more weight than that of the mixed column EHRS, RSMN; and therefore the 
lighter part of this column, that is, the solid body, will be carried upw'ard with a force equal to the dif¬ 
ference of the weights of the column of fluid EHMN, and the mixed column EHRS, RSMN. 

PROP. XLIV. A body which has the same specific gravity with the fluid, in which 
it is immersed, will remain suspended in any part of the fluid. 

The body RMNS being of the same specific gravity with the fluid, the column EHMN presses 
downward with the same force, whilst this body makes a part of it, as if the column consisted wholly of 
water, that is, with a force equal to the upward pressure against MN. Therefore the body RSMN, hav¬ 
ing its lower surface MN, and in like manner all its parts, pressed by equal forces in opposite directions 
will remain at rest. 

Exr. Let small glass images made hollow, and of specific gravity somewhat less than water, having a 
small orifice to receive water, swim in a large glass vessel nearly filled with water and covered over closely 
with a piece of bladder; by pressing the bladder with the hand, the air on the surface is compressed ; 
this pressure is communicated to the air in the images, which consequently receive a larger portion of 
water, and become in specific gravity as heavy as the water, or heavier, and either float in the water, 
or sink. 

PROP. XLV. A body specifically heavier than the fluid, in which it is immersed, may 
be supported in it by the upward pressure, if the pressure downward be taken away ; 
and a body specifically lighter than the fluid, in which it is immersed, will not rise in 
the fluid, if the upward pressure be taken away. 

For, in the first case, the pressure which the solid RSMN sustains from the weight of the fluid being 
removed, the solid may press downward with a force equal to, or less than, that of the column of fluid 
EHMN, that is, than that with which it is pressed upward, according to the degree of depth in the fluid 
at which the solid is placed. 

In the second case, as the upward pressure against MN is diminished, the downward pressure of the 
mixed column EHMN becomes equal to, or greater than, the upward pressure, and the solid will either 
float in the fluid, or sink. 

Exr. For the first part of the proposition, see Prop. III. Exp. 2 and 5. The second part may be thus 
confirmed. If a plane and smooth piece of hard wood, or of cork, be closely pressed down by the hand 
upon the plane and smooth bottom of any vessel, whilst mercury is pouring into the vessel; upon re¬ 
moving the pressure of the hand, the downward pressure of the mercury will prevent the wood from 
rising. 

PROP. XLVI. If a body float on the surface of a fluid specifically heavier than it¬ 
self, it will sink into the fluid till it has displaced a portion of. fluid equal in weight to 
the solid. 


Chap. IV. 


OF SPECIFIC GRAVITIES. 


63 


Let p t ei be a body, floating on a liquor specifically heavier than itself, it will sink into it till the Plate 5 * 
immersed part, rn ei , takes up the place of so much fluid as is equal to it in weight. For, in that case, 
e that part of the surface of the stratum upon which the' body rests, is pressed with the same degree 
.of force, as it would be, were the space rnei full of the fluid ; that is, all the parts of that stratum are 
pressed alike, and therefore the body, after having sunk so far into the fluid, is in equilibrio with it, and 
will remain at rest. 

Exp. 1. Place a cube of wood on a small jar, exactly filled with water ; a part of its bulk will be im¬ 
mersed, and will displace a quantity of the water; take the cube out of the water, and put it into a scale, with 
which an empty vessel in the other scale stands balanced; then pour water into that vessel till the equi¬ 
librium is restored ; that portion of water will fill up the jar in which the cube was placed. 

2. Let a glass jar with a weight sufficient to make it sink in water to about two thirds of its length, 
be placed first in a large vessel of water, and afterwards in one which is very little wider than the jar, 
and which has in it a small quantity of water, the jar will sink to the same depth in both vessels; that 
is, till so much of the vessel is under water as is equal in hulk to a quantity of the fluid whose weight 
is equal to that of the whole vessel. 

Cor. Hence arises a rule for estimating the specific gravities of fluids or solids. For, since (by this 
Prop.) the weights of the water displaced and of the solid are equal, their specific gravities are inverse¬ 
ly as their magnitudes; that is, the magnitude of the water displaced is to that of the solid, as the spe¬ 
cific gravity of the solid is to the specific gravity of the fluid; or (since the part immersed is equal in 
magnitude to the fluid displaced) the part immersed is to the whole, as the specific gravity of the solid 
to the specific gravity of the fluid. Consequently, the greater portion of any given solid is immersed in 
any fluid, the less is the specific gravity of the fluid ; and with respect to solids, inverting the proposi¬ 
tion, as the whole is to the part immersed, so is the specific gravity of the fluid to that of the solid ; 
whence, the greater is the portion of any solid immersed in a given fluid, the greater is its specific 
gravity. 

PROP. XLVII. A solid weighs less when immersed in a fluid than in open air, by 
the weight of a quantity of the fluid equal in bulk to the solid. 

If the body immersed were of the same specific gravity with the fluid, (by Prop. XLIV.) it would be 
supported in the fluid by the upward pressure. The fluid therefore sustains so much of the gravity of 
the bod}', or takes away so much of its weight, as is equal to the weight of that quantity of fluid which 
would fill the place taken up by the body. 

Or thus ; a body endeavours to descend by its whole weight; but (as was shown, Prop. XLI.) when 
it is immersed in a fluid, it is supported by a force equal to the weight of an equal bulk of that fluid. 

And since these two forces act in contrary directions, the wmight which the body retains in the fluid 
will be the difference between them; that is, it weighs as much less in the fluid than in the air, as the 
weight of a quantity of the fluid equal in bulk to the solid. 

Exp. Having provided a solid cylinder of lead which exactly fills a hollow cylinder of brass, place in 
one scale the hollow cylinder; under the same scale suspend by a string the solid cylinder, and balance 
the whole by weights; then immerse the solid cylinder in water, and the equilibrium will be restored 
by filling up the hollow cylinder. 

Remark. In strictness both the solid and fluid should be weighed in vacuo. The error, however, 
arising from the pressure of the air is very small, and may be neglected, unless where the body to be 
weighed is very light, and also where great precision is required. 

Cor. 1. Hence the specific gravities of different fluids may be compai’ed, by observing how much 
the same solid (specifically heavier than the fluids) loses of its weight in each fluid ; that fluid having 
the greatest specific gravity in which it loses most of its weight. 

Exp. Let a cubic inch of wood, made sufficiently heavy to sink in water, be immersed successively 
in different fluids ; it will displace a cubic inch of the fluid in which it is immersed; and since the cube 
(by Prop. XLVIII.) weighs less in the fluid, by the weight of a quantity of the fluid equal in bulk to the 
cube, its loss of weight will be the weight of a cubic inch of the fluid. 

Cor. 2. The weights which bodies lose in any fluid are proportional to their bulks. 

Exp. 1 Two balls of equal bulk, one of ivory, the other of lead, will lose equal weight in water. 

2. A piece of copper and a piece of gold being of equal weight in air, the gold outweighs the cop-* 
per in water. 

Cor. 3. If it be known what a cubic inch of any body loses in water, the solid content of any ir¬ 
regular mass of the same kind may be known, by observing how much more or less it loses, than a cu¬ 
bic inch would lose. 

Exp. Weigh a cubic inch and any irregular piece of wood of the same kind, and ohserve the difference 
of their weights. 


64 


OF HYDROSTATICS. 


Book III. 


Cor. 4. The weight of a solid body of the same specific gravity with the fluid, or of a portion of the 
fluid itself, suspended in the fluid, is not perceived, because this weight is supported, and not because 
the gravity of the body is lost or destroyed. 

PROP. B. If a and b be the specific gravities of two fluids which are to be mixed 
together; A and B their magnitudes, and c the specific gravity of the compound ; then 
A : B : : b — c : c — a , provided the magnitude of the compound be equal to the sum 
of the magnitudes of the parts when separate. 

Since (by Prop. A.) the weights of bodies are as their magnitudes, and specific gravities, conjointly, 
the weight of A = A X «, and that of B = B x b ; and the weight of the compound = A + B x c ; 
but the weight of the compound must be equal to the sum of the weights of the two parts, A -f B X c 
= A X « + B X ^ therefore A c -f- B c = A a -f- B 6, and Ac — A a = B 6 — Be; consequently A : B : : 
b — c: c — a. 

Exp. Let the specific gravity (to avoid fractions) of gold be 19, of silver 11, and of the compound 14 ; 
then the magnitude of the silver in the mixture is to that of the gold, as 19 — 14 : 14 — 11 : : 5 : 3. 

Cor. Hence the solution of that problem which was investigated by Archimedes, in order to detect 
the fraud ot the artist, who, instead of gold, was suspected of having substituted silver, in the crown of 
Hiero, king of Syracuse. If the proportion of the weights of each body is required, the ratios of their 
magnitudes, and of the specific gravities, must be taken conjointly ; in this case the weights of A and B 
are as a x b — c : b x c — a ; that is, the weight of the silver is to that of the gold, as 11 x 5 : 19 
X 3 : : 55 : 57. 

Schol. We easily deduce from this chapter the methods of obtaining the specific gravities of any 
bodies, taking rain water as a standard, a cubic foot of which being uniformly found to weigh 1000 
avoirdupois ounces. 

The weight which a body loses in a fluid is to its whole weight, as the specific gravity of the fluid 
is to that of the body; where three terms of the proportion being given, the fourth is easily found. 
Ex. If a guinea weigh in air 129 grains, and on being immersed in water lose 7^ of its weight, the pro¬ 
portion will be : 129 : : 1000 to the specific gravity of a guinea. By this method the specific gravi¬ 
ties of all bodies that sink in water may be found. 

Cor. 1. Hence if different bodies be weighed in the same fluid, their specific gravities will be as 
their whole weights directly, and the weights lost inversely. 

If a body to be examined consist of small fragments, they may be put into a small bucket and weigh¬ 
ed ; and then if from the weight of the bucket and body in the fluid, we- subtract the weight of the 
bucket in the fluid, there remains the weight of the body in the fluid. 

Cor. 2. If the same body be weighed in different fluids, the specific gravity of the fluids will be as 
the weights lost. 

Exp. The loss of weight sustained by a glass ball in water and milk is respectively 803 and 831 
grains; therefore the specific gravity of water is to that of milk as 803 : 831 ; that is, as 1.000 : 1.034. 
By the same method, the specific gravity of water is to that of spirits of wine, as 1.000 to .857. 

TABLE OF SPECIFIC GRAVITIES. 

Marble.2.705 

Green glass.2.600 

Flint ------ 2.570 

Ivory.1.825 

Sulphur.1.810 

Chalk.I.793 

Calculus humanus .... 1.542 

Lignum vitae - 1.327 

Coal .1.250 

Mahogany.1.063 

Milk .1.034 

Brazilwood - - - - - 1.031 

Box wood.1.030 

Rain water.1.000 

Ice..908 

Living men..891 

Ash..800 


Platina (pure) .... 23.000 

Fine gold - 19.640 

Standard gold - 18.888 

Mercury - 14.019 

Lead.11.325 

Fine silver - - - - -11.091 

Standard silver - - - 10.535 

Copper - 9.000 

Gun metal ----- 8.784 

Fine brass.8.350 

Steel.7.850 

Iron ------ 7.645 

Pewter ----- 7.471 

Cast iron.7.425 

Loadstone.4.930 

Diamond ----- 3.517 

White lead.3.160 






Chap. I. 


OF THE WEIGHT AND PRESSURE OF AIR. 


Maple..755 Fir..550 

Beech..700 Cork..240 

Elm..600 Common air..001-5^ 

Remark 1. The above table shows the specific weights of the various substances contained in it, and 
the absolute weight of a cubic foot of each body is ascertained in avoirdupois ounces, by multiplying 
the number opposite to it by 1000; thus the weight of a cubical foot of mercury is 14019 ounces 
avoirdupois, or 8761b. 

Remark 2. If the weight of a body be known in avoirdupois ounces, its weight in Troy ounces will 
be found by multiplying it into .91145. And if the weight be given in Troy ounces, it will be found in 
avoirdupois by multiplying it into 1.0971. 

Remark 3. Mr. Robertson, late librarian to the Royal Society, was the gentleman who investigated 
the specific gravity of living men, in order to know what quantity of timber would be sufficient to keep 
a manalloat in water ; supposing that most men were specifically heavier than river w’ater ; but the con¬ 
trary appeared to be the case from trials which he made upon ten different persons, whose mean spe¬ 
cific gravity was, as expressed in the table, 0.891, or about ^th less than common water.* Phil. Trans 
Vol. L. 

The scales made use of to determine the specific gravities of bodies, are called the hydrostatic bal¬ 
ance. 


BOOK III. PART II. 

OF PNEUMATICS. 

CHAPTER, I. 

Of the Weight and Pressure of the Air. 

Def. The Air or Atmosphere , is that fluid which encompasses the earth. 

PROP. XLVIII. The air has weight. 

This appears from experiment. 

Exp. 1. The air being exhausted, by an air-pump, from a glass receiver, the vessel will be held fast 
by the pressure of the external air. 

2. If a small receiver be placed under a larger, and both be exhausted, the larger will be held fast 
while the smaller will be easily moved. 

3. If the hand be placed upon a small open vessel, in such a manner as to close its upper orifice, it 
will be held down with great force. 

4. The upper orifice of an open receiver being closely covered with a piece of bladder, upon ex¬ 
hausting the receiver, the bladder will burst. 

5. In the same situation a thin plate of glass will be broken. 

6. Pour mercury into a wooden cup, closely placed upon the upper orifice of an open receiver • 
when the air beneath is exhausted, the pressure of the external air will force the mercury through the 
wood, and it will descend in a shower. 

7. On a transferrer let the air be exhausted from a long receiver; then let water be admitted through 
a pipe, by means of a cock; the water will rise in a jet d'eau. 

8. Fill a glass tube, about 3 feet long and closed at one end, with mercury; then insert the open 
end in a vessel of mercury; the mercury will remain suspended in the tube by the pressure of the ex¬ 
ternal air upon the surface of the mercury in the vessel; when this pressure is removed, by placing 
the tube and vessel under a receiver, and exhausting the air, the mercury will sink in the tube, and on 
re-admitting the air,'will- rise. 

9. If the same immersed tube be suspended from the beam of a balance, the weight necessary to 
counterpoise it, exclusive of the weight of the tube, is equal to that of the mercury sustained in the ba¬ 
rometer by the pressure of the atmosphere ; for the weight of the column of air incumbent upon the 
tube not being counterbalanced by the contrary pressure from below, which is employed in bearing up 
tbs mercury within the tube, must press upon the beam. 

* This must have been taken during inspiration, and the buoyancy of different individuals is probably very different. 

9 




66 


OF PNEUMATICS. 


Book III. Part II. 


10. Let a barometer tube, instead of being hermetically sealed at the top, be closely covered with a 
piece of bladder; the mercury wiU rise to the same height as in a common barometer; and on piercing 
the bladder with a needle, to admit the air, it will fall. 

Schol. Hence the pressure of the atmosphere on or near the surface of the earth is known ; the 
weight of any column of air being equal to the weight of the column of mercury, of the same diameter, 
supported in the barometer. And, since the height of this column varies with the weight of the atmos¬ 
phere, the varieties in the weight of the atmosphere are known by the barometer. 

11. Let the air be exhausted from a glass vessel, and by means of a cock let the vessel be kept ex¬ 
hausted; weigh the vessel whilst it is exhausted, and when the air is re-admitted; the difference is the 
weight of so much air as the vessel contains; which difference will be about 324 grains for a thousand 
cubic inches. 



PROP. XLIX. The air presses equally in all directions. 

Exp. 1. Let a bladder, tilled with air, be placed within a condensing receiver, the condensed air will 
make the bladder flaccid. 

2. In a tall phial let an orifice be made about 3 inches above the bottom ; stop this orifice ; through 
a cork in the neck of the phial insert a long tube open at each end ; and let its lower end be below the 
orifice in the side of the phial. The mouth of the phial being closed up about the tube, pour water 
into the tube till it is full. Upon opening the orifice, the water will be discharged till its surface in the 
tube is level with the orifice ; after which it vyill cease to flow, because the external lateral pressure of 
the air balances the perpendicular pressure upon the water in the tube. 

3. If a glass vessel be filled with water and covered with a loose piece of paper, on inverting the 
glass, the water will be kept from falling by the upward pressure of the air. 

4. If a vessel be perforated in small holes at the bottom, but closed at the top, the upward pressure 
of the air will keep the water within the vessel; as will appear by successively stopping and unstopping 
a small hole in the top of the vessel. 

5. Two brass hemispherical cups put close together, w 7 hen the air between them is exhausted, will 
be pressed together with considerable force. 

6. A syringe being fastened to a plate of lead, and the piston of the syringe being drawn upward 
with one hand, whilst the lead is held in the other, the air, by its upw r ard pressure, will drive back the 
syringe upon the piston; whereas if the loaded syringe be hung in a receiver, and the air be exhaust¬ 
ed, the syringe and lead will descend; but upon re-admitting the air, they will again be driven 
upward. 

7. If a thin glass vessel, whose aperture is closed, be placed under the receiver of an air pump, and 
the air exhausted from the receiver; the vessel will be broken by the pressure of air within. 

PROP. L. The pressure of the atmosphere varies at different altitudes. 

Exp. Put a glass tube, open at both ends, through a cork into a large phial containing a small quan¬ 
tity of coloured water; let the lower end of the tube be in the w ater; and let the cork and tube be 
closely cemented to the neck of the bottle. Then, blow through the tube, till the quantity of air within 
the phial is so increased, that the water will rise above the neck of the phial. Let this phial be placed 
in a vessel of sand, to keep the air within of the same temperature ; the water will stand at different 
heights in the tube, according to the elevation of the place where it is placed ; from whence it appears, 
that the pressure of the atmosphere varies at different altitudes. 

Cor. Hence the proportion of the specific gravity of air to that of water may be determined. If the 
difference in height of the two places w here the above experiment is made be 54 feet, and that differ¬ 
ence cause a difference of A of an inch in the height of the water ; it follows, that a column of water 
of -f of an inch, or of a foot, is equiponderant to a column of air of 54 feet having the same base ; 
therefore the gravity of water to that of air, is as 54 to -Jg-, or 864 to 1. In ascending the mountain of 
Snowdon in Wales, which is 3720 feet perpendicular height, it was found that the barometer sunk 3-^- 
inches. See Art. Barometer, Prop. LVII. 


PROP. LI. The force with which the wind strikes upon the sail of a ship, the ve¬ 
locity of the air and the dimension of the sail being given, will be as the square of the 
sine of the angle of incidence. 

Plate 5 Let represent the sail of a ship, with its edge toward the eye ; and let a circle be drawn upon 

y-jv iq[ the centre K ; whence K will be the middle of the sail, and AD its length. If the wind blow perpen¬ 
dicularly against the sail, all the air included within the space FADG will strike upon it. But if the 
sail is inclined in the position BE, all the air which strikes upon it, is included within the space HBEI. 



4 


Chap. II. 


OF THE ELASTICITY OF AIR. 


67 


If it were possible that the sail should be struck with the same quantity of air in the perpendicular 
position AD as in the oblique position BE, yet the quantity of the oblique stroke would he to the 
quantity of the direct stroke (by Book II. Prop. XVIII.) as the sine of incidence to the radius ; that is, 
since (supposing LK drawn parallel to BH, the direction of the wind, and BC perpendicular to KL) 
BKL is the angle of incidence, and BL its sine, as BL to AK. 

Again, if it were possible that the oblique stroke of the wind upon the sail BE should he equal to 
the direct stroke upon AD; yet, the column of air which strikes upon the sail directly, having AD for 
its base, and the column which strikes obliquely, having BC for its base, the quantity of air which strikes 
obliquely, is to that which strikes directly, as BC to AD, that is, as BL to AK ; but the velocities in either 
case are supposed to be the same ; therefore the momenta*, or forces with which the sails are struck, 
will be as the quantities of matter, that is, as BL the sine of incidence to AK the radius. 

Thus, the force with which the wind strikes the sail BC obliquely, is to the force with which it 
strikes an equal sail AD directly, as BL to AK on two accounts; first, because an oblique stroke is to a 
direct stroke in this ratio ; and secondly, because the quantity of air which strikes the oblique sail is to 
that which strikes the direct one in the same ratio. Consequently, upon both accounts together, the 
oblique force is to the direct one as BLxBL to AKxAK, or as the square of BL the sine of incidence 
to the square of AK the radius. But, the length of the sail, or AD being given, AK the radius is a 
given quantity. Therefore the force of the wind, in different obliquities of the sail, will be as BL* 
the square of the sine of incidence. 


CHAPTER II. 

Of the Elasticity of the Air. 

PROP. LII. The air is an elastic fluid, or capable of compression and expansion. 

Exp. 1. A blown bladder, pressed with the hand, will return into the form which it had before the 
pressure. 

2. A flaccid bladder, put under a receiver, when the external air is exhausted, becomes extended by 
the elasticity of the internal air. 

3. A bladder suspended within the' receiver, with a small weight hanging from it which touches the 
bottom, when the external air is exhausted, by the expansion of the internal air, will raise the weight. 

4. The bladder being put into a box, and a weight laid upon the lid, the weight, on exhausting the 
air, will be lifted up. 

5. If a tube, closed at one end, be inserted at its open end in a vessel of water, the fluid in the tube 
will not rise to the level of the water in the vessel, being resisted by the elastic force of the air within 
the tube. On this principle the diving-bell is formed. 

6. If a bladder be inclosed in a glass vessel so closely that the air in the vessel without the bladder 
cannot escape, but the air within the bladder communicates with the external air through the neck of 
the vessel; the external air being exhausted, the bladder will be closely pressed by the air in the ves¬ 
sel ; and when the air is re-admitted, the bladder will be distended. 

7. A shrivelled apple, under an exhausted receiver, will have its coat distended by the internal air. 

8. In the same situation, the air contained in a fresh egg will expel its contents from an orifice made 
in its smaller end. 

9. On green vegetables, and other substances, placed in a vessel of water under a receiver, whilst 
the air is exhausting, bubbles will be raised by the expansion of the internal air. 

10. Beer, a little warmed, will, from the same cause, whilst the internal air is exhausting, have 
the appearance of boiling. 

11. Let a cylindrical piece of wood (made just specifically heavier than water by fastening to it a 
small plate of lead) be placed in a vessel of water under a receiver; upon exhausting the air the wood 
will swim; some particles of air escaping from the wood, and hereby diminishing its specific gravity. 

12. Let a glass bulb, having a long neck, be put, with the neck downward, into a vessel of water; 
put the whole under a receiver, and exhaust the air; on re-admitting the air, that fluid, acting on the 
surface of the water in the vessel by its elasticity, will cause it to rise in the bulb, or, if the degree of 
exhaustion be great, nearly fill it. If the air be again exhausted from the receiver, the air remaining 
in the bulb, by its elasticity will expel the w ater from the bulb. 

13. Place a double transferrer upon the air-pump, with two receivers, exhaust one receiver; then 
open the pipe between the two receivers ; and the air in the unexhausted receiver will, by its elasticity 



OF PNEUMATICS. 


Book III. Part I). 


be in part driven into the exhausted receiver; and both receivers will have equal portions of air; but 
this air will be rarer in both than the external air; whence both the receivers will be held fast by the 
external pressure. 

PROP. LIII. The elastic spring of the air is equivalent to the force which com¬ 
presses it. 

If the spring with which the air endeavours to expand itself when it is compressed were less than 
the compressing force, it would yield still farther to that force; if it were greater, it would not have 
yielded so far. Therefore, when any force has compressed the air so that it remains at rest, the spring of 
the air arising from its elasticity can neither be greater nor less than this force, that is, must be equal to it. 

Exp. Let the air be exhausted from an open tube, whose lower part is inserted in a vessel contain¬ 
ing a small quantity of mercury, and let the air within the vessel be prevented from escaping; this air, 
by its elasticity, will force the mercury up the tube nearly to the height to which it would be raised by 
the pressure of the atmosphere. 

PROP. LIV. The space which any given quantity of air fills is inversely, and its 
density directly, as the force which compresses it. 

Exp. Let there be a bent tube of the form n k g, open at n and closed at g. Let a small portion of 
mercury be at the bottom k i. Then gi is tilled with air compressed by the weight of the atmosphere, 
equivalent to the weight of a column of mercury about 29£ inches in height. If more mercury be 
poured into the orifice n, the weight of this mercury is an addtional compressing force acting upon the 
airfg. Since (by Prop. V.) the columns of equal heights Ik, hi, balance each other, the air in the space 
g i is pressed both by the weight of the atmosphere and the column m l. If therefore ml be 29± inches, 
the air in gi is pressed with double the weight of the atmosphere, or with two atmospheres; and it 
will be found that it will be compressed into the space g h, half the space which the same quantity of air 
took up when it was pressed only with the weight of the atmosphere; therefore the space is inversely 
as the compressing force. And its density (Def. V.) is inversely as its bulk, or the space filled by it. 
Since therefore, both the compressing force and the density of the air are inversely as the space, the 
density must be directly as the compressing force. 

PROP. C. The density of the air being increased, the elasticity is increased in the 
same ratio. 

For (by Prop. LIII.) the elasticity is equivalent to the compressing force ; and (by Prop. LIV.) the 
compressing force is as the density ; therefore the elasticity is as the density. 

Exp. 1. Condense the air within a globular vessel, having a long neck, by blowing through the 
neck, the increased elasticity of the air within the vessel will force out water. 

2. The glass bulb and vessel, used in Experiment 12, Prop. LII. being placed within a condensing 
receiver, and the quantity of air in the receiver increased, w ater will rise into the bulb. 

3. The quicksilver in the gauge of the condenser will be forced upward in the tube by increasing 
the density of the air. 

4. Condense the air in different degrees in the condenser, and observe the gauge; and note the 
different heights at which a column of mercury is supported by air of different degrees of density. 

PROP. LV. The air consists of particles which repel each other with forces which 
are inversely as the distances between their centres. 

An elastic fluid equally compressed in all directions must have all its particles at equal distances 
from each other; for if the distances are unequal, where it is the least, the repelling force will be the 
greatest, and the particles will move toward the side where there is less repulsion, till the forces be¬ 
come equal, that is, till the particles are equally distant, or the fluid becomes every where of the same den¬ 
sity. Suppose, then, two equal cubes of air, A and B ; it is manifest, from the nature of the cube, that the 
number of particles in the whole mass A is equal to the cube of the number of particles in the line cl e • 
and, in like manner, that the number of particles in the mass B is equal to the cube of the number of 
particles in the line h i. And the density of these two equal cubes of air A and B will be as the number 
of particles contained in them. Therefore the density of the cube A is to the density of the cube B as 
the cube of the number of particles in the line d e to the cube of the number of particles in the line h i. 
But, since these lines d e, h i, are of a given length, the number of particles in each will be greater, as 
the distances between their centres are less, that is, will be inversely as those distances. Whence, the 
cube of the number of particles in de, h i , will be inversely as the cube of the distance between their 


Chap. II. 


OF THE ELASTICITY OF AIR. 


69 


centres. And it has been shown, that the density of the mass A is to the density of the mass B, as the 
cube of the number of particles in de to the cube of the number of particles in hi. Therefore the 
density of A is to the density of B inversely, as the cube of the distance between the centres of the 
particles. 

Also, in compressing any mass, A, every surface, as def g, is pressed closer to the surface next beyond 
it. And the repulsion of the surface d cf g against the surface next beyond it will be (all other circum¬ 
stances being equal) as the number of repelling particles in that surface, that is, as the square of the 
number of particles in the line d e. But the number of particles in the line de is inversely as the dis¬ 
tance between their centres. Therefore the square of the number of particles in de , that is, the number 
of repelling particles in the surface def g, that is, the repulsion of this surface against the next beyond it, 
is inversely as the square of the distance between the particles. Again, where the number of particles 
in each surface is given, if it be supposed that the particles repel each other with a force which is 
inversely as the distance between their centres, since the surfaces are at the same distance from each 
other with the particles which compose them, the repulsion of the surfaces must be in the same ratio. 
Thus, the repulsion in the mass A is to that in the mass B, inversely as the distances of the particles, if 
only their approach to each other be considered. And it has been shown that the repulsion is inversely as 
the square of these distances, if only the number of particles be considered. Therefore on both accounts 
taken together, the repulsion is inversely as the cube of the distance of the particles. And (by Prop. 
LII1.) the compression is as the repulsion; therefore the compression is inversely as the cube of the 
distance of the particles. 

Now it was shown above, that the density of A is to the density of B inversely as the cube of the 
distance of the particles. Therefore, when a fluid consists of particles which repel each other with 
forces inversely as the distances between the cenires of the particles, the density of the fluid will be as 
the compressing force. But it was shown (Prop. L1V.) that the density of the air is as the compressing 
force. Therefore the air consists of particles which repel each other with forces which are in¬ 
versely as the distances between their centres. 

Schol. From the doctrine of the elasticity of the air, the phenomena of sound may be explained. 

When the parts of an elastic body are put into a tremulous motion, by percussion, or the like, as 
long as the tremors continue, so long is the air included in the pores of that body, and likewise that 
which presses upon its surface, affected with the like tremors and agitations. Now, the particles of air 
being so far compressed together by the weight of the incumbent atmosphere as their repulsive forces 
permit, it follows, that those which are immediately agitated by the reciprocal motions of the particles 
of the elastic body, will, in their approach toward those which lie next them, impel these also toward each 
Other, and hereby cause them to be more condensed than they were by the weight of the incumbent 
atmosphere, and in their return will suffer them to expand themselves again; hence the like tremors and 
agitations will be propagated to them ; and so on, till having arrived at a certain distance from the 
body, the vibrations cease, being gradually destroyed by a continual successive propagation of motion 
to fresh particles of air throughout their progress. 

Thus it is that sound is communicated from a tremulous body to the organ of hearing. Each vi¬ 
bration of the particles of the sounding body is successively propagated to the particles of the air, till it 
reaches those which are contiguous to the tympanum of the ear (a fine membrane distended across it), 
and these particles, in performing their vibrations, impinge upon the tympanum, which agitates the air 
included within it; which being put into a like tremulous motion, affects the auditory nerve, and thus 
excites in the mind the sensation or idea of what we call sound. 

Now, since the repulsive force of each particle of air is equally diffused around it every way, it 
follows, that when any one approaches a number of others, it not only repels those which lie before it 
in a right line, but the rest laterally, according to their respective situations; that is, it makes them 
recede every way from itself as from a centre. And this being true of every particle, the tremors will 
be propagated from the sounding body in all directions, as from a centre; and further, if they are con¬ 
fined for some time from spreading themselves by passing through a tube, will, when they have passed 
through it, spread themselves from the end in every direction. In like manner, those which pass 
through a hole in an obstacle, they meet in their way, will afterward spread themselves from thence, 
as if that was the place where they began ; so that the sound will be heard in any situation whatever, 
that is not at too great a distance. 

The utmost distance, at which sound of any kind has been heard, is about .200 miles, which, is said 
to have been observed in the war between England and Holland in the year 1672. The watch words 
All’s well , given at New Gibraltar, was heard at the Old, a distance of 12 miles. In both these cases, 
the sound passed over water, which with respect to conducting sound, is of the greatest consequence. 
By an experiment made on the river Thames, a person was distinctly heard to read at the distance of 140 


70 


OF PNEUMATICS. 


Book III. Part II. 


feet on the water, on land at that of 76; in the latter case no noise intervened, but in the former there 
was some occasioned by the flowing of the water against the boats. Watermen observe, that when the 
water is still, the weather calm, and no noise intervenes, a whisper may be heard across the river. 
After water, stone may be reckoned the best conductor of sound. Brick has nearly the same proper¬ 
ties as stone. 

Since the repulsive force with which the particles of air act upon each other, is reciprocally as their 
distances (by Prop. LV.) it follows, that when any particle is removed out of its place by the tremors of 
a sounding body, or the vibrations of those which are contiguous to it, it will be driven back again by 
the repulsive force of those toward which it is impelled, with a velocity proportional to the distance 
from its proper place, because the velocity will be as the repelling force. The consequence of this is, 
that, let the distance be great or small, it will return to its place in the same time; for the time a body 
takes up in moving from place to place will always be the same, whilst the velocity it moves with is pro¬ 
portional to the distance between the places. The time therefore in which each vibration of the air is 
performed, depends on the degree of repulsion in its particles, and so long as that is not altered, will be 
the same at all distances from the tremulous body; consequently, as the motion of sound is owing to the 
successive propagation of the tremors of a sounding- body through the air, and as that propagation de¬ 
pends on the time each tremor is performed in, it follows, that the velocity of- sound varies as the elas¬ 
ticity of the air, but continues the same at all distances from the sounding body. 

The velocity of sound, according to Mr. Derham, is at the rate of 1142 feet in a second of time. 
Hence, with a stop watch, may be easily estimated the distance of thunder, for by multiplying the num¬ 
ber of seconds between the flash and clap of thunder by 1142, the distance is given in feet. Or thus, 
persons in good health have about 75 pulsations at the wrist in a minute, consequently in 75 pulsations, 
sound flies about 13 miles, that is, one mile in about six pulsations. Example. On seeing the flash of a 
gun at sea, 54 pulsations at the wrist were counted before the report was heal'd, consequently the dis¬ 
tance of the ship is ^ = 9 miles. 

Moreover, since the undulatory motion of the air, which constitutes sound, is propagated in all direc¬ 
tions from the sounding body; it will frequently happen, that the air, in performing its vibrations, will 
impinge against various objects, which will reflect it back, and so cause new vibrations the contrary 
way; now, if the objects are so situated, as to reflect a sufficient number of vibrations back to the same 
place, the sound will be there repeated, and is called an echo. And, the greater the distance of the ob¬ 
jects is, the longer will be the time before the repetition is heard. And when the sound in its progress 
meets with objects, at different distances, sufficient to produce an echo, the same sound will be repeated 
several times successively, according to the different distances of those objects from the sounding body ; 
which makes what is called a repeated echo. Echoes repeat more by night than in the day. 

If the vibrations of the tremulous body are propagated through a long tube, they will be continually 
reverberated from the sides of the tube into its axis, and by that means prevented from spreading, till 
they get out of it; whereby they will be exceedingly increased, and the sound rendered much louder 
than it would otherwise be, as in the speaking trumpet. 

The difference of musical tones depends on the different number of vibrations communicated to the 
air, in a given time, by the ti'emors of the sounding body; and the quicker the succession of the vibra¬ 
tions is, the acuter is the tone, and the reverse. 

PROP. LVI. The elasticity of air is increased by heat. 

Exp. To the bottom of a hollow glass ball let an open bended tube be affixed. Let the lower part 
of the bended tube and part of the ball be filled with mercury; the external surface will be pressed by 
the weight of the atmosphere ; and the intei'nal surface will be equally pressed by the spring of the air 
inclosed within the vessel. If the ball be immersed in boiling water, the increased elasticity of the in¬ 
cluded air will raise the mercury in the small tube. The same may be shown by immersing in boiling 
water a tube, closed at one end, into which a small quantity of mercury has been admitted, inclosing a 
portion of air within the tube. 

Schol. 1. The wind is no other than the motion of the air upon the surface of the globe. The prin¬ 
cipal cause of the wind is, that the atmosphere is heated over one part of the earth moi’e than over another. 
For, in this case, the warmer air being rarefied, becomes specifically lighter than the rest; it is therefore 
overpoised by it and raised upward, the upper parts of it diffusing themselves every way over the top 
of the atmosphere; while the neighbouring inferior air rushes in from all parts at the bottom ; which it 
continues to do, till the equilibrium is restored. Upon this principle it is, that most of the winds may 
be accounted for. 

Under the Equator , the wind is always observed to blow from the east point. For, supposing the sun 
to continue vei'tical over some one place, the air will be more rarefied there ; and consequently, the 
neighbouring air will rush in from every quarter with equal force. But, as the sun is continually shift- 


Chap. II. 


OF THE ELASTICITY OF AIR. 


71 


ing to the westward, the part where the air is most rarified, is carried the same way; and therefore the 
tendency of all the lower air, taken together, is greater that way, than any other. Thus the tendency 
of the air toward the west becomes general, and its parts impelling one another, and continuing to move 
till the next return of the sun, so much of its motion, as was lost by his absence, is again restored, and 
therefore the easterly wind becomes perpetual. 

On each side of the Equator , to about the thirtieth degree of latitude, the wind is found to vary 
from the east point, so as to become north-east on the northern side, and south-east on the southern. 

The reason of which is, that as the equatorial parts are hotter than any other, both the northern and 
southern air ought to have a tendency that way ; the northern current, therefore, meeting in this pas¬ 
sage with the eastern, produces a north-east wind on that side ; as the southern current, joining with 
the same, on the other side the Equator , forms a south-east wind there. 

This is to be understood of open seas, and of such parts of them as are distant from the land ; for 
near the shores where the neighbouring air is much rarefied, by the reflection of the sun’s heat from 
the land, it frequently happens otherwise; particularly on the Guinea coast, the wind always sets in 
upon the land, blowing Avesterly instead of easterly. This is because the deserts of Africa lying near 
the Equator , and being a very sandy soil, reflect a greater degree of heat into the air above them ; 
which being thus rendered lighter than that which is over the sea, the wind continually rushes in upon 
the land to restore the equilibrium. 

That part of the ocean which is called the Rains , is attended with perpetual calms, the wind scarce¬ 
ly blowing sensibly either one way or another. For this tract being placed between the westerly wind 
blowing from the ocean toward the coast of Guinea , and the easterly wind blowing from the same coast 
to the westward thereof, the air stands in equilibrio between both, and its gravity is so much diminished 
thereby, that it is not able to support the vapour it contains, but lets it fall in continual rain, from 
whence this part of the ocean has its name. 

There is a species of winds, observable in some places within the Tropics , called by the sailors Mon¬ 
soons, or Trade Winds, which during six months of the year, blow one way ; and the remaining six the 
contrary. The occasion of them in general is this; when the sun approaches the northern Tropic , 
there are several countries, as Arabia , Persia , India , &,c. which become hotter, and reflect more heat 
than the seas beyond the Equator , which the sun has left; the winds therefore, instead of blowing from 
thence to the parts under the Equator , blow the contrary way; and when the sun leaves those countries, 
and draws near the other Tropic , the winds turn about, and blow on the opposite point of the compass. 

From the solution of the general trade winds, we may see the reason, why, in the Atlantic ocean, a 
little on this side the thirtieth degree of north latitude, there is generally a west, or south-west wind. 

For, as the inferior air, within the limits of those winds, is constantly rushing toward the Equator , from 
the north-east point, or nearly so, the superior air moves the contrary way; and therefore, after it has 
reached these limits, and meets with air, that has little or no tendency to any one point more than to 
another, it will determine it to move in the same direction with itself. 

In our own climate we frequently experience, in calm weather, gentle breezes blowing from the 
sea to the land, in the heat of the day ; which phenomenon is very agreeable to the principle laid down 
above; for the inferior air over the land being rarefied by the beams of the sun, reflected from its sur¬ 
face, more than that which impends over the water, the latter is constantly moving on to the shore, in 
order to restore the equilibrium, when not disturbed by stronger winds from another quarter. 

From what has been observed, nothing is more easy than to see, why the northern and southern 
parts of the world, beyond the limits of the trade winds, are subject to such variety of winds. For the 
air, upon account of the lesser influence of the sun in those parts, being undetermined to move toward 
anv fixed point, is continually shifting from place to place, in order to restore the equilibrium, when¬ 
ever it is destroyed, by the heat of the sun, the rising of vapours or exhalations, the melting of snow 
upon the mountains, or other circumstances. 

Exp. Fill a large dish with cold water ; into the middle of this put a water plate filled with warm water. 

The first will represent the ocean; and the other an island rarefying the air above it. Blow out a wax 
candle, and if the place be still, on applying it successively to every side of the dish, the smoke will be 
seen to move toward the plate. Again, if the ambient water be warmed, and the plate filled with cold 
water, let the smoking wick of the candle be held over the plate, and the contrary will happen. 

Schol. 2. Heat expands all bodies, solid as well as fluid. 

Exp. 1, 2. Water may be rarefied into steam, and will become exceedingly elastic, acting with great 
power, as in the eolipile, and in steam engines. See art. X. Prop. LVII. 

3. Metals expand by heat, and the degrees of their expansion are measured by the Pyrometer, 
which is an instrument invented to render the smallest expansions sensible. 

Various machines have been contrived for this purpose, by Ferguson, Dessaguliers, De Luc, &ic. p] ate jo., 
but the general principle may be thus illustrated. Let abc be a lever, whose fulcrum is b , acting upon Fig. 13.- 


OF PNEUMATICS. 


Book III. Part II. 


r.i 


0 > 




Plate 5. 
Fig. 23. 


another lever cde, whose fulcrum is d; this again acts upon a third lever e/g, whose fulcrum is /, and 
let x be a metallic rod, one end of which rests against an immoveable obstacle P, and the other end 
against the lever a b c, at a. If a lamp be put under this rod, the heat will increase its length, and put 
the levers in motion. Now by the principle of the lever, 

Vel. of a : Vel. of c : : ab : be 

Vel. of c : Vel. of e : : c d : d e 

Vel. of c : Vel. ofg : : ef : f g. 

Therefore Vel. of a : Vel. of g :: ab x cd x ef : be x de xfg- 

Hence, if a b , cd, e f be small in proportion to b c, d e,fg, a trifling increase in the length x will pro¬ 
duce a very considerable motion in the point g , which may be measured upon the graduated arc y z. 

Ex. If a 6, c d, e j\ be each equal to 1, and b c, d e,fg , each equal to 15, then if the rod increase 
but the 3375th part of an inch, the point g will describe 1 inch; consequently by dividing each inch in 
the graduated arc into 20 parts, an expansion in the rod of less than a 60 thousandth part of an inch 
easily becomes visible. 

Mr. Ferguson found the expansion of metals to be in the following proportion ; iron and steel 3 ; 
copper 4|; brass 5 ; tin 6 ; and lead 7. An iron rod 3 feet long, is about ~ of an inch longer in sum¬ 
mer than in winter. See Ferguson’s first Lecture and Supplement; Dessagulier’s Exp. Phil. Cham¬ 
ber’s Cyclopaedia, by Dr. Rees. 

4. Mercury expanding or contracting by an increase or decrease of heat in the air, is made the 
measure of heat in thermometers. See Art. VIII. Prop. LVII. 

Schol. 3. It is found by experiment, that air is necessary to the existence of sound, of animal life, of 
fire, and of explosion. 

Exp. 1. Let a bell ring under an exhausted receiver, and in a condenser. 

2. Let a lighted candle be extinguished under a receiver. 

3. Let gunpowder fall upon red hot iron placed within an exhausted receiver. 

Schol. 4. The elasticity of the air affords a method of determining the depth of the sea where a 
line cannot be used. 

A wine glass immersed in water with its mouth downward will not become filled, because the spring 
of air will prevent the water from entering beyond a certain point. The diving bell is constructed on 
this principle. 

PROP. LVII: To explain the nature and use of sundry Hydraulic and Pneumatic 
Instruments. 

1. The Syphon. 

Let DEC be a bended tube, having one leg longer than the other. This instrument, used for draw¬ 
ing off liquors, is called the syphon. If the shorter leg of the tube be inserted in a vessel of fluid, and 
if by sucking with the mouth a vacuum be produced in the tube, or if the tube be filled with the fluid 
before it is used, the fluid will run oft' from the vessel. The cause of which may be thus explained; 
the orifice C, of the longer leg, is exposed to the pressure of the atmosphere ; also, since the fluid 
within the shorter leg is supported by the surrounding fluid in the vessel, the pressure upon the orifice 
D is that of the atmosphere. The two equal orifices are then acted upon by equal pressures; the dif¬ 
ference of the lengths of the columns of atmosphere being too small to cause any perceptible difference 
in their pressure. But these equal pressures are counteracted by the pressures of two unequal columns 
of fluid ED, EC. If, therefore, the pressures of the columns of atmosphere be more than sufficient to 
balance those of the columns of fluid, that which acts with the lesser force, that is, the lesser column 
DE, is more pressed against the column CE, than the column CE is pressed against DE at the vertex 
E. Consequently, the column EC must yield to the greater pressure, and flow off through the ori¬ 
fice C. 

Exp. 1. Draw off water by a syphon. 

2. Whilst mercury is passing off'from a vessel by a syphon, let the air be exhausted from the vessel, 
and the fluid will cease to run. 

3. Intermitting fountains are natural syphons. 

II. The Syrin GE. 

Let a hollow cylindrical tube have a small orifice at one end; at the other end insert a solid cylin¬ 
der so exactly fitted to the tube, that no air can pass along its sides, and fix a handle to the solid cylin¬ 
der. If that end of this instrument which has the smaller orifice, be inserted in water, and the solid 
cylinder or piston be drawn back, a vacuum will be produced within the syringe ; and the pressure of 


Chap. II. 


OF THE ELASTICITY OF AIR. 


73 


the atmosphere on the surface of the water, meeting with no opposite pressure, will force the water 
into the tube, from whence it may be forcibly expelled, by pushing down the piston. 

III. The Common Pump. 

In this useful instrument, a handle, acting upon a pin as a lever of the first kind, draws up a piston 
AD, fitted to the shaft or barrel of the pump, as described in the syringe. This piston has a hole, over 
which is a valve of leather, loaded with lead, opening upward. Toward the lower part of the shaft, is 
inserted a plug C, which also has in it a hole, and a valve which opens upward. When the piston, or 
sucker, is drawn up from the plug, a vacuum is produced in the shaft between D and C, into which the 
air contained in the lower part of the pipe expands itself. By repeated strokes the air escapes through 
the upper valve, and the vacuum becomes so perfect, that the external air, pressing without counterac¬ 
tion upon the surface of the water, in the well or reservoir in which the shaft is supposed to be in¬ 
serted, forces the water through the valves at C and D, into the space AD ; from whence it is prevent¬ 
ed from returning downward, by the valves, which are closely pressed down by the incumbent fluid. 
If therefore the handle be repeatedly lifted up, the column of water will increase upon every stroke, 
till it rises to the level of the spout, and is discharged. But if the height be more than 34 feet, the 
water cannot be raised; for such a column is equal to the weight of a column of the atmosphere of the 
same diameter. 

IV. The Forcing Pump. 

In this pump, the piston is one entire cylinder, .as in the syringe. The water is raised into the 
pipe between A and D, as in the common pump; from hence it is forced, by the downward pressure of 
the piston, or forcer, through a tube inserted in the side of the main shaft. In this side-tube a valve is 
inserted at E to prevent the water from returning, and when a sufficient quantity is raised, it is dis¬ 
charged by the spout. 

The common engine for extinguishing fires consists of two such forcing pumps, which convey the 
water into a reservoir made air tight, into which a pipe is inserted. As this reservoir fills with wafer, 
the air within it is proportionally condensed, and therefore forces the water up a cylinder from which 
it is conveyed, at pleasure, by leathern pipes. 

V. The Condenser. 

This instrument, which is used to force air into any vessel, is a syringe, having a solid piston, and 
a valve in the lower part of its barrel which opens downward. By thrusting down the piston, the air 
is forced through the valve, which is afterward held close by the elasticity of the condensed air. When 
the piston is lifted up, a vacuum is produced, till it is raised above a small hole in the barrel, when the 
air rushes in, and is again discharged through the valve. 

Artificial Fountains are formed by the help of a condenser, which throws any quantity of air into 
a vessel in part filled with water; which, by its elasticity, forces the water up into pipes from which 
it is conveyed at pleasure. 

The Air-Gun is an instrument in the form of a gun, by which a quantity of condensed air is sud¬ 
denly set free, and drives a ball through the barrel with great force. 

VI. The Air-Pump. 

This instrument, the use of which is to exhaust the air from any vessel, has two strong barrels A, A, 
which communicate with a cavity in D ; within each of which near the bottom, is fixed a valve open¬ 
ing upward, and two pistons, one in each barrel, having a valve which likewise opens upward. These 
pistons are moved by means of a cog-wheel in the piece TT, to the axis of which the handle B is fixed, 
and whose teeth catch in the racks of the pistons CC, and move them upward or downward. PQR is a 
circular brass plate, having at Its centre the orifice K of a concealed pipe that communicates with the 
cavity. In the piece D at V is a screw that closes the orifice of another pipe, for the purpose of ad¬ 
mitting the external air when required. Upon VV is placed the short barometer guage'for the purpose 
of showing the degrees of exhaustion. When the handle is turned one of the pistons is raised, and a 
vacuum produced in its barrel. By means of the pipe, which passes from the orifice K in the 
plate upon which the receiver LAI, or vessel to be exhausted, stands, to the part of the barrel beneath 
the lower valve, the air contained in the receiver, communicating with the barrel, raises the lower 
valve by its elastic spring, and expands into the vacuum. Thus a part of the air in the receiver is ex¬ 
tracted. By turning the handle the contrary way, the same etfect is produced in the other barrel ; 
whilst, the first piston being depressed, the air which had passed from the receiver is compressed, and 

10 


Plate 5. 
Fig. 21 


Plate 5. 
Fig. 22i 


Plate 12. 
Fig. 12. 


74 


OF PNEUMATICS. 


Book III. Part II. 


escapes through a valve in the piston. This operation is continued till the air is nearly exhausted 
from the receiver; for it can never be perfectly exhausted, since at each stroke only such a part of the 
air which remained is taken away, as is to the quantity before the stroke, as the capacity of the barrel, 
to that of the receiver, pipe, and barrel taken together; which may be easily proved in the following 
manner. 

Let R = the content of the receiver and pipe, B = the content of the barrel. 

If L = the quantity of air in R before the stroke, and / = the quantity exhausted by it; and since, 
the piston being raised, the air is uniformly diffused through R and B, and that in B extracted by the 
stroke, consequently L : / :: R -f B : B, or l : L :: B : R -f B ; that is, the quantity of air extracted is to 
the quantity before the stroke, as the capacity of the barrel is to that of the receiver, pipe, and barrel 
taken together. 

Cor. 1. Let L, M, N, &,c. be the quantities of air, before any successive strokes; /, m, n , &.c. the 
quantities-exhausted by each stroke ; and L : l :: R + B : B :: M : m : : N : », &c. by division L : L —/ 
(M) : : M : M —m (N) : : N : N — n (O), &c. or L : M :: M : N :: N : O, &c. therefore L, M, N, &c. are 
in a decreasing geometric progression, whose common ratio is that of R + B : R. If R = 2B, then li -f* 


B : R :: 3 : 2, and M 




&c. and the quantities of air are equal to L, \ L, | L, 3 8 7 L, &c. 


Cor. 2. Since L : M :: l : m ; M : N :: m : n, &c. /, m, n, &c. are in a decreasing geometric pro¬ 
gression, whose common ratio is that of R -f- B : R. If, as in the last Cor. Ii -j- B : R : : 3:2, then m 

^ l QTtb • 

= —, n = &.c. and the quantities of air exhausted by the successive strokes are /, -| /, | l : -£j /, 

3 * 3 a / 


&c. 

Cor. 3. If R be to B in any finite ratio as 3 : 2, the receiver can never be perfectly exhausted by 
any finite number of turns; for let the number of turns be n, and Q. the last remainder, then Q = L 
X f l n > supposing L to be the quantity of air in the receiver at first; and this quantity L x §| n is finite, 
since n is finite. 


Plate 12. 
Fig. 4 . 


Plate 12. 
Fig. 3. 


VII. The Barometer. 

(1) If a glass globe be exhausted of air, and balanced at one end of a beam, upon admitting the air 
the globe preponderates. This experiment not only, in common with others beforementioned, shows 
that the air has weight, but also what that weight is. The density of air was found, by Mr. Hauksbee, 
to be 885 times less than that of water, when the barometer stood at 2 9$ inches. Hence as a cubic inch 
of water weighs 253.18 grains Troy, a cubic inch of air weighs 0.286 grains. And if mercury be 14 
times heavier than water, the specific gravity of air is to that of mercurj' as 1 to 885 x 14 = 12390. 

(2) If a glass tube AB, about 32 or 33 inches long, hermetically sealed at one end, be filled with 
mercury, and then inverted into a bason D of the same fluid, the mercury in the tube will stand at an 
altitude above the surface of that in the bason between 28 and 31 inches. A tube thus filled, and grad¬ 
uated from 28 to 31 inches, is called a barometer. The height of the mercury in the tube above the 
surface of the mercury in the bason is called the standard altitude, which, in this country, fluctuates be¬ 
tween 28 and 31 inches; and the difference, between the greatest and least altitudes, is called the scale 
of variation. 

Now the mercury in the barometer tube will subside, till the column be equivalent to the weight of 
the external air upon the surface of the mercury in the bason, and is therefore a true criterion to meas¬ 
ure that weight, and chiefly directed to that purpose, in order to foretell the changes in the weather. 

If each inch of the scale of variation AD, (fig. 5, made larger for the sake of perspicuity) be divid¬ 
ed into ten equal parts, marked 1, 2, 3, increasing upward, and a vernier LM, whose length is I^ths °f 
an inch, be likewise divided into ten equal parts, increasing downward, and so placed as to slide 
along the graduated scale of the barometer, the altitude of the mercury in the tube, above the surface 
of that in the bason may be found, in inches and hundredth parts of an inch, by this process. If the sur¬ 
face E of the mercury in the tube do not coincide with a division in the scale of variation, place the 
index of the vernier M even with this surface, and observing where a division of the vernier coincides 
with one of the scale, the figure in the vernier will show what hundredth parts of an inch are to be 
added to the tenths immediately below the index. If the surface of the mercury be between 6 and 7 
tenths above 30 inches, and, the index of the vernier being placed even with it, the figure 8 upon 
the vernier coincide with a division upon the scale, the altitude of the barometer will be 30 inches 
and xItt an inch. For each division of the vernier being greater than that of the scale by °f an 
inch, (for the tenth part of a tenth of an inch is the hundredth part of an inch) and there being eight 
divisions, the whole must be T |^ of an inch above the number 6 in the scale, and the height of the 
mercury is therefore 30.68 inches. 


Chap. II. 


OF THE ELASTICITY OF AIR. 


75 


Cor. 1 . Hence, if the atmosphere were homogeneous, its altitude would be easily found. For by the 
former part of this article, when the mercury stood at 29§ inches, the density of the air was to that of 
mercury as 1 to 12390; consequently the altitude of a homogeneous atmosphere would be equal to 
12390 x 29£ = 5.77 miles. The real height of the atmosphere may be determined from the beginning 
and end of twilight. See Hook VII. Prop. XXXIX. 

Cor. 2. The barometer has been applied to the measuring of the heights of towers, mountains, &c. 
Since 12390 inches of air, near the surface of the earth, is equal to one inch of mercu^; 1239 inches, 
or about 103 feet of air, must be equal to of an inch of mercury. Therefore if a barometer be car¬ 
ried up any great eminence, the mercury will descend y 1 - of an inch for every 103 feet that the barom¬ 
eter ascends. This corollary supposes that the atmosphere near the surface of the earth is every 
where of the same density, which is so far from being true, that the conclusions drawn from the suppo¬ 
sition deviate from fact even in small altitudes, as appears from the following observations made by Dr. 
Nettleton. 


Town of Halifax 

Perpen. Altitudes. 
102 

Lowest Station. 
29.78 

Highest Station. 
29.66 

Diff. 

0.12 

Coal mine 

236 

29.50 

29.23 

0.27 

Halifax-hill 

507 

30.00 

29.45 

0.55 


See Abr. Phil. Trans. Vol. vi. 

M. De Luc, Sir George Shuckburgh, and General Roy, have considered this subject very attentive¬ 
ly, and have laid down certain rules, which, with proper corrections, on account of the difference of the 
temperature of the air, will hold good for all altitudes within our reach. See De Luc on the Modifica¬ 
tions of the Atmosphere. Phil. Trans. Vol. lxxvii. 

Cor. 3. When the mercury in the barometer stands at the altitude of 30 inches, the pressure of the 
air upon every square inch is rather more than 151b. avoirdupois. Now, supposing the surface of a 
middle-sized man to be 14| square feet, the pressure upon him, when the air is lightest, will be 13.2 
tons, and when heaviest, it will be 14.3 tons, the difference of which is 24641b. The difference of 
pressure must affect us in regard to our health and animal spirits, especially when the change takes 
place suddenly. 

For a description of the different kinds of barometers, see Parkinson’s Hydrostatics, p.97. 

VIII. The Thermometer. 

The thermometer is an instrument calculated for measuring the temperature of the air, and other 
bodies contiguous to it, as to heat and cold, being usually a cylindrical glass tube, containing air, water, 
oil, spirits of wine, mercury, &c. which fluids are found to occupy different portions of the tube in differ¬ 
ent temperatures, and these portions being measured, exhibit the different expansions of the included 
fluid. 

AB represents a glass tube, whose end A is blown into a bulb ; this bulb and part of the tube being plate 12. 
filled with quicksilver, the least change of the bulk of quicksilver, and consequently of the temperature Fig. 2. 
of contiguous bodies, is shown by the rise or fall of the surface in the tube, which is indicated by the 
scale a b affixed to the frame of the instrument. 

The thermometer chiefly used in Great Britain, is that constructed by Fahrenheit; in which there 
are 180 divisions between the freezing and boiling water points, the freezing point being reckoned 32° 
above zero, or the commencement of the scale; consequently the boiling water point is 212°.* 

A good thermometer must possess the following properties; the capacity of the tube should be very 
small and regular, and its upper end must be hermetically sealed. The empty space must be as free as 
possible from air. The scale must be well adjusted, and accurately divided according to the capacity 
of the tube. Thermometers with small bulbs, and tubes in proportion, are the most to be depended 
upon, for a large volume of mercury is not sufficiently sensible to the change of temperature. 

Since the thermometers of Fahrenheit and Reaumur are those most in us$, it will be often found 
convenient to be able readily to convert the degrees on Fahrenheit’s scale into those of Reaumur, and 
vice versa; and as one degree on Reaumur’s scale is equal to 2.25°, or to |° of Fahrenheit; and as 
the former scale places the freezing point at zero, and the latter places it at 32; the following canons 
will reduce the degrees on the one to the corresponding one9 on the other. 

F — 32 

1 . To convert the degrees of Fahrenheit to those of Reaumur; —- -x 4 = R; thus the 167° 

of Fahrenheit answers to the 60° of Reaumur. 

* The scale on Reaumur’s thermometer, which is principally used on the continent, begins at the freezing point, and 
proceeds both ways, from 0 or zero. From freezing to boiling water are 80 degrees. For the construction, uses, &c. of 
this and several other thermometers, see Parkinson’s Hydrostatics, p. 154—169. 




76 


OF PNEUMATICS. 


Book III. Part II. 




2. To convert the degrees of Reaumur into those of Fahrenheit; —--(- 32 = F. 

Thus the 40° of Reaumur answers to the 122° of Fahrenheit. See No. 4, Appendix to Lavoisier's 
Chemistry. 

R is evident, that the thermometers hitherto described, are limited in their extent. The mercuri¬ 
al thermometer extends no farther than the heat of boiling mercury, which answers to 600° of Fahren¬ 
heit’s scale ; but the heat of solid bodies in the state of ignition exceeds that of boiling mercury. To 
remedy this defect, Mr. Wedgewood has contrived a thermometer for measuring the higher degrees of 
heat, by means of a distinguished property of argillaceous bodies, viz. the diminution of their bulk by 
fire. This diminution commences in a dull red heat, and proceeds regularly as the heat increases, till 
the clay becomes vitrified. This thermometer, therefore, marks with precision, the different degrees 
of ignition from the red heat visible only in the dark, to the heat of an air furnace. Its construction is 
extremely simple. It consists of two rulers fixed upon a flat plane, a little farther asunder at one end 
than at the other, leaving an open longitudinal space between them. Small pieces of allum and clay, 
mixed together, are made of such a size as just to enter at the wide end; they are then heated in the 
fire along with the body whose heat we wish to determine. The fire according to the degree of heat 
it contains, contracts the earthy body, so that applied to the wide end of the guage, it will slide on 
toward the narrow end, less or more, according to the degree of heat to which it has been exposed. 
Each degree of Mr. Wedgewood’s thermometer answers to 130 degrees of Fahrenheit; and the scale 
begins from 1077 of Fahrenheit. Hence the following 

TABLE. 


Extremity of Wedgewood’s scale 

Cast iron melts - 

Least welding heat of iron - 

Fine gold melts - 

Fine silver melts - 

Brass melts - 

Red heat fully visible in day light 

Red heat fully visible in the dark 

Mercury Boils, also expressed oils 

Lead melts - - - 

Bismuth melts - 

Tin melts - ... 

Nitrous acid boils 

Cow’s milk boils ... 

Water Boils - 

Heat of the human body 

Oil of olives begins to congeal 

Water freezes and snow melts - 

Milk freezes - - 

Urine and vinegar freeze 

Strong wine freezes 

A mixture of snow and salt freezes 

Mercury freezes - 


Fahrenheit’s Wedgewood’s 

scale. scale. 

32277° .... 240° 

21877 - - - 160 

12777 .... 90 

5237 .... 32 

4717 - 28 

3807 - - - 21 

1077 .... o 

947 - 1 

600 

540 ( Note. If these three metals be mixed together by fu- 
460 < sion, in the proportion of 5, 8. and 3, the mixture 
408 ( will melt in a heat below boiling w r ater. 

242 
- 213 

212 

92 to 99 
43 
32 
30 
28 
20 

0 to 4 
— 39 or 40 


Cold produced at Hudson’s Bay, by a mixture of 

vitriolic acid and snow - - - - — 69 


IX. The Hygrometer. 


The hygrometer is an instrument for measuring the degrees of moisture in the air ; of which there 
are various kinds ; for whatever contracts and expands by the moisture and dryness of the atmosphere, 
is capable of being formed into a hygrometer. Such are most kinds of wood; catgut, twisted cord, 
the beard of wild oat, &c. The following are very simple in their construction, and will serve to 
explain the principle of the instrument. 

Plate 12, L Stretch a catgut or a common cord, ABD, along a wall, passing it over a pulley B ; fixing it at 
Fig. 6. one end A, and to the other hanging a weight E, carrying a small index F. Against the same wall, fit a 
metal plate HI, divided into any number of equal parts, and the hygrometer is complete. 



Chap. II. 


OF THE ELASTICITY OF AIR. 


77 


For it is known, that moisture sensibly shortens catgut, cord, &,c. and that as the moisture evapo¬ 
rates, they return to their former length. Hence the weight E, with the index, will ascend when the 
air is moist, and descend when it becomes drier; and the divisions on the scale will show the degrees 
of moisture or dryness. This hygrometer may be made more sensible and accurate by straining the 
catgut over several pullies placed in a parallel or any other position. 

2. The sponge hygrometer is constructed as follows; BC is the beam of a balance; to the end B is Plate 12. 
hung a piece of sponge, so cut as to contain as large a superficies as possible, which must be exactly F*g- 7. 
balanced on the other side by another thread of silk D, on which is strung some very small leaden shot 
at equal distances, so adjusted as to cause an index E to point to G, the middle of the graduated arc 
FGH, (made large for distinction’s sake) when the air is in a middle state between the greatest moisture 
and the greatest dryness. Under this silk, strung with shot, is placed a shelf I, for that part of the shot 
to rest upon which is not suspended. When the moisture imbibed by the sponge increases its weight, 
it will raise the index, and vice versa when the air is dry. 

To prepare the sponge, it may be proper first to wash it in water very clean, and when dry again, 
dip it in water or vinegar in which there has been dissolved sal ammoniac, or salt of tartar; after which 
let it dry again. Salt of tartar, or any other salt, or pot-ashes, may be put into the scale of a balance, 
and used instead of the sponge. 

X. Steam Engine. 

The steam engine is a machine which derives its moving power from the elasticity and condensa¬ 
tion of the steam of boiling water. The high importance of this machine to the mechanical arts of life, 
especially where immense powers are required, has given birth to many considerable improvements 
both in its construction and mode of operation. 

The following is a description of one of the earliest steam engines, which, as it exhibits the general 
principles of the machine, will be deemed sufficient in a work only introductory to science. A history 
of the steam engine, from its first construction by Capt. Savary, down to the present time, in which are 
included all the great improvements made by the ingenious Mr. Watt of Birmingham, will be found in 
the Encyclopaedia Britannica, Vol. xvii. Part n. 

H represents the boiler on its furnace ; E the cylindrical vessel of iron, in which the piston 00 riate 12. 
moves up and down. The cavity between the piston and bottom of the cylinder is made air tight. F F»g-11. 
is a cock to admit the steam into the cylinder. IK is a lever, attached to the piston at I, and at K to 
the piston of the pump which works on that side. PQ is a solid piston moving in the pipe RM, and 
loaded with a heavy weight at P. ABC is the main pipe that receives the water forced from RM 
through a valve at C, opening outw r ard. N is an air vessel communicating with the main pipe. At D 
is a valve opening upward, and at M is the water to be raised. 

The engine is represented at the end of a forcing stroke, which is likewise its position when at rest. 

Suppose the main pipe ABC-to be filled with water, and the water in the boiler H to boil strongly. 

The cock F being opened, the steam rushes into the cylinder, and being much lighter than air, rises to 
the top, and expels the air through a valve in the bottom of the cylinder. F is then shut, and the cock 
G communicating with the main pipe is opened, which, by spouting cold water against the bottom of the 
piston, condenses the steam. A vacuum being thus obtained, the pressure of the atmosphere forces 
the piston down to the bottom of the cylinder; the lever IIv is moved, and the piston PQ, with its 
weight is raised, and the water ascends in the pipe MR upon the principle of the common pump. The 
cock G being now shut, and F opened, the steam enters the cylinder, and counteracts the pressure of the 
atmosphere on the piston 00 ; consequently, the weight P prevails, and drives down the piston PQ, 
forcing the water through the valve C into the main pipe and its air vessel. The use of the air vessel 
is to prevent the main pipe from bursting by the sudden entrance of the water; for the air at N being 
elastic, gives way to the stroke, and its reaction during the the time of elevating the piston PQ con¬ 
tinues the motion of the water, so that its velocity is no more than half what it would have been if it 
had been impelled by starts, and rested during the raising of the piston. By opening the cock G, and 
shutting F, (which is done by a single operation) the steam is again condensed, the pressure of the at¬ 
mosphere again prevails, and thus the work may be continued at pleasure. * 

The power of some of the steam engines, construced by Messrs. Boulton and Watt, is thus described 
as taken from actual experiment. An engine, having a cylinder of 31 inches in diameter, and making 
17 double strokes per minute, performs the work of 40 horses, working night and day, (for which three 
relays, or 120 horses must be kept) and burns 11,000 pounds of Staffordshire coal per day. A cylinder 
of 19 inches, making 25 strokes of 4 feet each per minute, performs the work ot 12 horses, working 
constantly, and burns 3,700 pounds of coal per day. These engines will raise more than 20,000 cubic 
feet of water, 24 feet high, for every hundred weight of good pit coal consumed by them. 


OF PNEUMATICS. Book III. Part II. 

XI. The Hydrometer. 

The Hydrometer, an instrument usually applied to find the specific gravities of liquids, is thus con¬ 
structed ; AB is a hollow cylindrical tube of glass, ivory, copper, &c. joined to a hollow ball D, at the 
bottom of which is a smaller ball E, containing some quicksilver, or shot, by which the instrument is so 
poised, that it swims vertically in a liquid. The stem AB is graduated in such a manner, that the fig¬ 
ures exhibit the magnitudes of the parts below, and consequently, the specific gravities of the different 
fluids in which it descends to those figures. Thus if the parts immersed in water , and spirits of wine , 
be as 10 to 11.1, then the specific gravity of the water will be to that of the spirits of wine as 
11.1 to 10. 

To make this instrument of more service, there has been added a little plate, or dish, at the top of 
the tube, upon which may be placed weights, as convenience may require. For example; if the 
whole instrument float, immersed in spirits to a certain point, it will require an additional weight to 
sink it to the same depth in water. Suppose the instrument to weigh 10 dwts. and to be adjusted to 
rectified spirits ot wine, it will then require an additional weight of 1.6 dwt. to sink it to the same 
point in water. Consequently, the specific gravity of water is to that of rectified spirits of wine as 
11.6 to 10, or as 10 to 8.6. 


78 

Plate 12. 






BOOK IT 


OF MAGNETISM 

Def. I. The earth contains a mineral substance which attracts iron, steel, and all 
ferruginous substances ; this is called a natural magnet. 

Def. II. The same substance has the power to communicate its properties to all 
ferruginous bodies ; those bodies, after having acquired the magnetical properties, 
are called artificial magnets. 

Those magnets are also made without the assistance of the natural magnet, as will 
hereafter be shown. 

Schol. The property of attraction in the magnet was that by which it was first discovered. Every 
substance that contains iron, is more or less attracted by the magnet; and so universally is this truly 
important metal disseminated, that there are very few substances which are not in some degree capa¬ 
ble of being attracted by the magnet. In this way iron is found to enter into the composition of ani¬ 
mals, vegetables, minerals, and even into that of the atmosphere. On this subject, see Cavallo on Mag¬ 
netism, Chap. vi. Part I. 

Def. III. Those points in a magnet, which seem to possess the greatest power, are 
called the poles of the magnet. 

Def. IV. The magnetical meridian is a vertical circle in the heavens, which 
intersects the horizon in the points to which the magnetical needle, when at rest, is 
directed. 

Def. V. The axis of a magnet is a right line, which passes from one pole to 
the other. 

Def. VI. The equator of a magnet is a line perpendicular to the axis, and exactly 
between the two poles. 

Schol. The distinguishing and characteristic properties of a magnet, are, (1.) Its attractive and re¬ 
pulsive powers. (2.) The force by which it places itself, when freely suspended, in a certain direction 
toward the poles of the earth. (3.) Its dip or inclination toward a point below the horizon. (4.) The 
property which it possesses of communicating the foregoing powers to iron and steel. 

Def. VII. The direction of the dipping needle in any place is called the magnet - 
ical line. 

PROP. I. That mineral substance which is called the loadstone, or magnet, has 
the property of attracting iron ; but no other body whatever, unless it has a mixture 
of iron. 

Exp. 1 . The action of the magnet on iron may be shown on needles, steel filings, &c. 

2. Let a needle be suspended from a loadstone, and a string passing through its eye be fastened 
to the beam of a balance placed under it; the degree of force with which it is attracted, may be 
measured. 



80 


OF MAGNETISM. 


Book IV. 


Schol. Some philosophers have supposed that iron is not the only substance attracted by the magnet, 
Mr. Kirwan says, that nickel , when purified from iron, becomes more, instead of less magnetic, and ac¬ 
quires the properties of a magnet. Mr. Cavallo instituted a number of experiments, with a view of as¬ 
certaining whether any other bodies than ferruginous ones, were attracted by the magnet. After all, 
he does not decide positively on the question. 

PROP. II. The action, and reaction of the magnetic power, are mutual and equal. 

A piece of iron, or steel, or other ferruginous substance, being brought within a certain distance of 
one of the poles of a magnet, is attracted by it, so as to adhere to the magnet, and not to suffer itself to 
be separated without an evident effort. Tiffs attraction is also mutual, for the iron attracts the mag¬ 
net, as much as the magnet attracts the iron; since if they be placed on pieces of wood or cork, so as 
to float upon the surface of water, it will be found that the iron advances toward the magnet, as well as 
the magnet advances toward the iron ; or, if the iron be kept steady, the magnet will move toward it. 

Schol. The strength of magnetic attraction varies according to different circumstances; such as, the 
strength of the magnet;—the weight and shape of the body presented to it;—the magnetic, or unmag- 
netic state of that body ; the distance between it and the magnet, &c. 

The attraction is strongest near the surface of the magnet, and diminishes as it recedes from it; the 
law of this diminution has not yet been ascertained. 

The four following experiments, accurately made by Professor Musschenbroek, will exhibit some of 
the irregularities respecting magnetic attraction. In these experiments, the magnet was suspended to 
one scale of an accurate balance, and under it there was successively placed on a table at different dis¬ 
tances, another magnet, or piece of iron; and at each distance, the degree of attraction between 
the iron and the magnet was ascertained by weights put into the other scale. The results were 
as follow; 


Exp. 1. In this experi¬ 
ment a cylindrical mag¬ 
net, weighing 16 drams, 
was suspended to the 
scale ; and on the table 
a piece of iron of the 
same shape and weight. 


Exp. 2. A spherical 
magnet, of the same di¬ 
ameter as the last, but of 
greater strength, was af¬ 
fixed to the scale, and the 
cylindrical magnet used in 
the preceding experiment 
was placed on the table. 


Exp. 3. Instead of the 
cylindrical magnet, the 
cylinder of iron was plac¬ 
ed on the table, and un¬ 
der the globular magnet. 


Dist. in 
inches. 
6 

5 - 
4 

3 - 

2 

1 - 
0 


Attract, 
in grains. 

- 3 



6 

9 


- 18 
- 57 


Dist. in 
inches. 
6 

5 - 
4 

3 - 

2 

1 - 

0 


Attract, 
in grains. 
- 21 

- 27 

- 34 

- 44 

- 64 

- 100 
260 


Dist. in 
inches. 

6 - 
5 - 
4 - 

3 - 
2 - 
1 - 
0 - 


Attract, 
in grains. 

- 7 

- 91 

- 15 

- 25 

- 45 

- 92 
340 


Exp. 4. A globe of iron 
of the same diameter as 
the magnet, was now plac¬ 
ed on the table. 


Dist. in 
inches. 

8 - 
7 - 
6 - 
5 - 
4 - 

3 - 
2 - 
1 - 
0 - 


Attract. 


in grains. 


- 1 
- 2 

- 3- 

- 6 

- 9 
- 16 

- 30 

- 64 

290 


Cor. It appears from the second and third experiments, that, when in contact, a magnet attracts 
another magnet with less force than it does a piece of iron. This has been confirmed by many other 
experiments. But the attraction between the two magnets begins from a greater distance than between 
the magnet and the iron; hence it must follow a different law of decrement.* 


PROP. III. The attraction and repulsion of magnetism is not sensibly affected by 
the interposition of bodies of any sort except those which are ferruginous. 

Exp. 1. Suppose a magnet placed at an inch distance from a piece of iron requires an ounce of force 
to remove it, or, which is the same thing, suppose the attraction toward each other is equal to one 
ounce; it will be found that the same degree of attraction remains constantly unaltered, though a plate 
of other metal, glass, paper, &c. be interposed between the magnet and the iron, or though they be in¬ 
closed in separate boxes of glass or other matter. 


* Mr. Coulomb has ascertained that the force of both magnetic and electric influence, like gravitation is inversely As 
the square of the distance. ’ 3 









Book IV. 


OF MAGNETISM. 


81 


2. Move steel filings placed on a brass plate, in water, &c. by holding a magnet under the vessel. 

3. Sprinkle steel dust on a sheet of paper, under which is placed a magnet, or two magnets having 
their poles opposite to each other, and at the distance of about an inch. 

4. A needle under an exhausted receiver will be attracted at the same distance, as in the open air. 

Schol. 1. Heat weakens the power of a magnet; and a white heat destroys it entirely. Hence it 

appears, from this cause alone, besides others which may concur, the power of a magnet must be con¬ 
tinually varying. 

Schol. 2. The attractive power of a magnet may be increased considerably by gradually adding 
more weight to it; for it is found that a magnet will keep suspended on one day a little more weight 
than it did the preceding; which additional weight being added to it, on the following day it will 
be found that the magnet can keep suspended a weight still greater, and so on, as far as to a certain 
limit. 

On the contrary, by putting a very small weight of iron to it, the magnet may gradually lose much 
of its strength. 

Schol. 3. Among natural magnets, the smallest generally possess a greater attractive power, in pro¬ 
portion to their size, than those which are larger. There have been natural magnets not exceeding 
20 or 30 grains, which would lift a piece of iron that weighed 40 or 50 times more than themselves. 

A small magnet, worn by Sir. I. Newton in a ring, weighing but about 3 grains, is said to have taken 
up 746 grains, or nearly 250 times its own weight. And Mr. Cavallo has seen one of 6 or 7 grains’ 
weight, which w r as capable of lifting a weight of 300 grains. But magnets of two pounds and upwards, 
seldom lift up ten times their own weight of iron. 

PROP. IV. The magnetic power may be communicated from the loadstone; and 
from one piece of iron to another, which then becomes an artificial magnet; and this 
communication of power is without apparent loss of power in the loadstone. 

Exp. 1. Take a bar of soft iron, about three feet long and one inch thick, (some kitchen pokers will 
answer for this experiment) and place it upright, or rather in the magnetical line. Then present a 
magnetic needle to the various parts of the bar from top to bottom, and the lower half of the bar will 
be possessed of the north polarity, capable of repelling the north, and of attracting the south pole of 
the needle, and the upper half is possessed of the south polarity. The attraction is strongest at the 
very extremities of the bar, it diminishes as it recedes from them, and vanishes about its middle point. 

If the bar is turned upside down, the south pole will become north, and the north will become the 
south pole. In the southern parts of the globe, the lower part is a south pole; or more generally, the 
extremities of the bar will acquire the polarities corresponding to the nearest poles of the earth. 

If an iron bar be left a long time in the direction of the magnetic line, or even in a perpendicular 
posture, it will sometimes acquire a great magnetic power. Tongs, pokers, &c. by being often heated, 
and set to cool again in an erect posture, frequently acquire a considerable magnetic virtue. Magnet¬ 
ism is often communicated to iron and steel by repeated blows of the hammer; by some experiments of 
Mr. Cavallo, it appears that this effect is often produced on brass, hence it is necessary carefully to 
examine the brass before it is used in the construction of theodolites, &c. 

2. Place two magnets A and B in a right line, so that the north end of the one is opposed to the Plate 13. 
south end of the other, and at such a distance that the bar to be touched may rest upon them. Take Fig. 1. 
now two other bars, D and E, and apply the north end of D,* and the south end of E to the middle of 

the untouched bar C, elevating their other ends so as to make an acute angle with the said bar. 

Separate the bars D and E, drawing them different ways along the surface of C, but preserving the 
same elevation ; then removing the bars D and E to the distance of a foot or more from the bar C, and 
bringing the north and south ends into contact, apply them again to the middle of C. This 
process being repeated several times to each surface of the bar C, it will be found to have acquired a 
strong and permanent magnetism. 

3. Take twelve bars, six of soft steel, and six of hard, the former T;o be each three inches long, A cf 
an inch broad, of an inch thick ; with two pieces of iron, each half the length of one of the bars, 
but of the same breadth and thickness. The 6 hard bars to be each inches long, 4- an inch broad, 
and ? 3 o of an inch thick, w ith two pieces of iron of half the length but of the same breadth and thick¬ 
ness of one of the hard bars ; and let all the bars be marked with a line quite round them at one end ; 
then take an iron poker and tongs, or two bars of iron, the larger they are, and the longer they have 
been used, the better; and fixing the poker upright, or rather in the magnetical line, between the 
knees, hold to it near the top, one of the soft bars, having its marked end downward, by a piece of 

* The north ends of magnetic bars are generally marked with a cross'or straight line, as are also the north ends of the 
hofse-shoe, or any other shaped magnets. 

II 


82 


OF MAGNETISM. 


Book IV. 


Plate 13. 
Fig. 2. 


Plate 13, 
Fig. 3. 


sewing silk, which must be pulled tight by the left hand that the bar may not slide; then grasping the 
tongs with the right hand, a little below the middle, and holding them nearly in a vertical position, let 
the bar be stroked by the lower end from the bottom to the top about 10 times on each side, which 
will give it a magnetic power sufficient to lift a small kej' at the marked end ; which end, if the bar 
were suspended on a point, would turn toward the north, and is therefore called the north pole ; and 
the unmarked end, for the same reason, is called the south pole. Four of the soft bars being impreg¬ 
nated after this manner, lay the other two parallel to each other, at a quarter of an inch distance, be¬ 
tween the two pieces of iron belonging to them, a north and a south pole against each piece of iron ; 
then take two of the four bars already made magnetical, and place them together so as to make a double 
bar in thickness, the north pole of one even with the south pole of the other; and the remaining 
two being put to these, one on each side, so as to have two north, and two south poles together, sepa¬ 
rate the north from the south poles at one end by the interposition of some hard substance I, and place 
them perpendicularly with that end downward on the middle of one of the parallel bars AC, the two 
north poles toward its south end, and the two south poles toward its north end. Slide them three or 
four times backward and forward the whole length of the bar ; then removing them from the middle of 
this bar, place them on the middle of the other bar BD as before directed, and go over that in the 
same manner; then turn both the bars the other side upward, and repeat the former operation ; this 
being done, take the two from between the pieces of iron; and placing the two outermost of the touch¬ 
ing bars in their stead, let the other two be the outermost of the four to touch these with ; and this 
process being repeated till each pair of bars have been touched three or four times over, will 
give them a considerable magnetic power. 

When the small bars have been thus rendered magnetic, in order to communicate the magnetism to 
the large bars, lay two of them on the table, between their iron conductors, as before ; then form a 
compound magnet with the six small bars, placing three of them with the north poles downward, and 
the three others with the south poles downward. Place these two parcels at an angle, as was done 
with four of them, the north extremity of the one parcel being put contiguous to the south extremity 
of the other, and, with this compound magnet, stroke four of the large bars, one after another about 
twenty times on each side, by which means they will acquire some magnetic power. 

When the four large bars have been so far rendered magnetic, the small bars are laid aside, and the 
large ones are strengthened by themselves, in the same manner as was done with the small bars. 

To expedite the operation, the bars ought to be fixed in a groove, or between brass pins, otherwise 
the attraction and friction between the bars will be continually deranging them, when placed between 
the conductors. 

This whole process may be gone through in about i an hour, and each of the large bars, if well 
hardened, will lift about 28 ounces Troy, and they are fitted for all the purposes of magnetism, in navi¬ 
gation and experimental philosophy. The half dozen being put into a case in such a manner that no 
two poles of the same name may be together, and their irons with them as one bar, they will retain the 
virtue they have received ; but if their power should, by making experiments, be ever so much im¬ 
paired, it may be restored without any foreign assistance in a few minutes. 

This method of communicating magnetism was sent to the Royal Society by Mr. Canton, in the 
year 1751. 

Schol. 1 . The magnetic virtue may be readily communicated by the horse-shoe magnet, much in 
the same way as in the preceding experiment. 

Schol. 2. A small compass needle may be touched by being put between the opposite poles of 
two magnetic bars. Whilst it is receiving the magnetism, it will be violently agitated, moving back¬ 
ward and forward as if it were animated ; and when it has received as much magnetism as it can ac¬ 
quire in this way, it becomes quiescent. 

Another method of communicating magnetism to a compass needle, is by means of the combined 
horse-shoe magnet, from the centre of which draw that half of the needle which is to have the contra¬ 
ry pole ; from a considerable distance draw the needle over it again. This repeated twenty times or 
more, and the same for the other half, will sufficiently communicate the power. 

PROP. V. Two magnets having a free motion will attract when different poles 
are directed toward each other, and repel when the adjacent poles are of the same 
name. 

Exp. A needle turning on its centre will be attracted or repelled by another, as different, or the 
same poles are brought near to each other. 

Schol. 1 . If the magnetic powers are very unequal, or the two bodies are forcibly brought together, 
they will attract with the same poles. 


Book IV. 


OF MAGNETISM. 


83 


Exp. 1 . Suspend a magnet by a thread, and let a small needle be brought near it, making poles of 
the same name contiguous. 

2. Bring two very unequal needles into contact at the same poles, suspended in the same manner, 
they will cohere. 

Schoi.. 2. The following experiments will show the attraction of the magnet on ferruginous bodies 
which arc not magnetic. 

Properly speaking, however, the magnet has no action upon unmagnetic bodies, for any ferruginous 
body becomes magnetic, on being presented to the magnet, and then is attracted by it. 

Exp. 1 . Place a magnetic needle upon a pin stuck on a table, and when it stands steady, place an 
iron bar 8 inches long, and \ an inch thick upon the table, so that one end of it may be on one side 
of the north pole of the needle, and near enough to draw it a little out of its natural direction. In this 
situation approach gradually the north pole of a magnet to the other extremity of the bar, and you will 
see that the needle’s north end will recede from the bar in proportion as the magnet is brought nearer 
to the bar. 

The reason of this phenomenon is, that, by the approach of the north pole to the magnet, in the 
first case, the extremity of the iron bar next to it acquires a south polarity, and consequently, the 
opposite extremity acquires a north polarity, by which the needle is repelled ; but in the second case, 
when the north pole of the magnet is brought near the bar, the end of the bar next to it acquires a 
south polarity, and the opposite end, acquiring the north polarity, causes the north end of the needle 
to recede. 

2. Tie two pieces of soft iron wire, AB, AB, each to a separate thread AC, which join at top, and Plate 13. 
suspend them on a pin so that the wires may hang at some distance from the wall. Then bring the Fig. 5. 
marked end D of a magnetic bar just under them, and it will be seen that the wires repel each other 

more or less in proportion to the distance of the magnet. The same may be shown by means of the 
south pole of the magnet. 

If the wires be of soft iron, they will, on removing the magnet, soon collapse; but if steel wires, 
or two sewing needles be used, they will retain their magnetic virtue, and continue to repel each other. 

3. Take four pieces of steel wire, or four common sewing needles, tie threads to them, and join 
them two and two, as in the last experiment; then bring the same pole of the magnet under both pairs, 
by which means they will acquire a permanent magnetism, and the wires of each pair will repel each 
other. After putting the magnet aside, bring one pair of the wires near the other pair, so that their 
lower extremities may be level, and the four wires will repel each other, and form a kind of square. 

4. Strew some iron filings upon a sheet of paper laid on a table, and place a small artificial magnet 
among them, then give a few gentle knocks to the table with the hand, so as to shake the filings, and 

they will dispose themselves round the bar in the manner represented by the figure; many particles p )ate 13 
clinging to one another, and forming themselves into lines, which at the very poles, are in the same fi°\ g. 
direction with the axis of the magnet; a little sideways of the poles they begin to bend, and then they 
form complete arches, reaching from a point in the north half of the magnet to a point in the other 
half which is possessed of the south polarity. 

5. Tie a thread to one end of a bit of soft iron wire AB, about four inches long, and suspend it p late 
freely; let a bar of soft iron CD be so supported, as to have one of its extremities C about $ of an inch Fig. 7. 
distant from the lower extremity B of the wire. Bring now either pole of a strong magnet EF under 

it, and the end B of the wire will recede from C, because they are both possessed of the same polarity; 
but if the magnet be applied to the upper part of the wire, in the situation GH, then the end B of the 
wire will be attracted by the extremity C of the iron bar, because, supposing G to be the north pole 
of the magnet, C acquires a south polarity, and attracts the end B ; because B being farthest from the 
north pole G, acquires also the north polarity. 

Schoi.. 3. Hence methods are easily devised to ascertain whether a body possesses any magnetism, 
and in case it does, to find out the poles. 

Exp. 1 . To ascertain whether a body has any attraction toward the magnet . 

If the body contain an evident quantity of iron, it will be perceived as soon as it is brought in con¬ 
tact with the magnet, as a certain force will be required to separate them. 

If the body be not sensibly attracted by the magnet in this way; let it be placed by means of a 
piece of cork or wood, upon some water, or mercury, in a common soup plate, in which situation let 
a magnet be brought sideways to it, and the attraction will be manifest by the body coming toward the 
magnet. 

2. To ascertain whether a given body has any magnetism. 

The only difference in this experiment from the last is, that instead of a magnet, must be used a 
piece of soft clean iron, about one inch long, and of half an ounce in weight. , 

3. A magnetic body being given , to find out its poles. 


I 


Book IV. 


84 OF MAGNETISM. 

Present the various parts of the surface of the magnetical body successively to one of the poles of a 
magnetic needle, and the parts possessed of a contrary polarity will be discovered by the needle’s stand¬ 
ing perpendicularly towards them. Then present the various parts of the surface of the same body to 
the other pole of the needle. 

Def. VIII. There is a point between the two poles where the magnet has no at¬ 
traction nor repulsion ; this point is called the magnetic centre , though it is not always 
exactly between the poles. 

PROP. VI. If a magnet be cut through the middle, or any way broken in two, 
each piece will become a complete magnet, and the parts which were contiguous will 
become opposite poles. 

Plate 13. Exp. 1. Take a magnetic bar AB, six or eight inches long, and £ of an inch thick, having only two 

F 'g- 8- poles A and B. The magnetical centre of this bar will be in, or very near, its middle C. Now if, 

by a smart stroke from the hammer, part of the magnet be broken off as FB, it will be found that 
the part of the fragment contiguous to the fracture has acquired the contrary polarity, and a magnetical 
centre B will be generated. 

At first the magnetic centre of this fragment is nearer to the fracture F, but in time it advances 
toward the middle of the fragment. The original centre C of AF, after the fracture, will likewise ad¬ 
vance nearer to the middle of it. 

2. A steel bar, of the same size as that mentioned in the last experiment, being made quite hard, 

may be broken into two parts, and so pressed together as to appear whole. In this situation it may 

be rendered magnetic by the application of very powerful magnets to its extremities ; and the whole 
bar will be found to have two poles at its extremities, and one magnetic centre in its middle ; but if 
the parts be separated, each will be found to have two poles and a magnetic centre. 

Cor. Hence it is seen that the magnetic centre may be removed ;—it may be removed also, by 
striking a magnetic bar, by heating it, by hard rubbing, &c. 

PROP. VII. Magnetism requires some time to penetrate through iron. 

Exp. Place a bulky piece of iron weighing 40 or 50 pounds, so near a magnetic needle as to draw 
it a little out of its direction, apply one of the poles of a strong magnet to the other extremity of the 
iron, and several seconds will be required before the needle can be affected by it. The interval is 
greater or less according to the size of the iron and the strength of the magnet. 

Def. IX. A magnet is said to be armed , when its poles are surrounded with plates 
of iron or steel. 

PROP. VIII. A magnet will take up much more iron when armed , than it can 
alone. 

As both magnetic poles together attract a much greater weight than a single one, and as the two 
poles of a magnet are generally in opposite parts of its surface, in which situation the same piece of 

iron cannot be adapted to them both at the same time ; therefore it has been common, to place two broad 

pieces of softiron to the poles of a magnet, and projecting on one side, because in that case, the pieces of 
iron being rendered magnetic, another piece of iron could be conveniently adapted to their projections, 
so that both poles may act at the same time. Those pieces of iron, called the armature , are generally 
held fast upon the magnet by means of a silver or brass box. Thus AB represents the magnet CD, CD 
Plate 13 represents the armature or pieces of iron, the projections of which are DD, and to which the piece of 

Fig. 4. iron F made to adhere. The dotted lines represent the brass box, having a ring E at top by which 

the armed magnet may be suspended. In this manner the two poles of the magnet, which are at A and 
B, are made to act at DD. 

For this purpose, and to avoid armature, artificial magnets have been constructed in the shape of a 
horse-shoe, having their poles in the truncated extremities. 

PROP. IX. A magnetical needle, accurately balanced on a pivot or centre, will 
settle in a certain direction, either duly, or nearly north and south, called the inagnet- 
ical meridian. 

This is known by long experience. 


Book IV. 


OF MAGNETISM. 


The directive power of the magnet is the most wonderful and useful part of the subject. By it 
mariners are enabled to conduct their vessels through vast oceans in any given direction ; by it miners 
are guided in their works below the surface of the earth; and travellers conducted through deserts 
otherwise impassable. 

The usual method is to have an artificial magnet suspended, so as to move freely, which will always 
place itself in or near the plane of the meridian north and south; then by looking upon the direction of 
this magnet the course is to be directed so as to make any required angle with it. Thus, suppose a 
vessel setting off from any place in order to go to another which is due west of the former; in that 
case, the vessel must be so directed that its course may be always at right angles with the situation of 
the magnetic needle, the north end of which must be to the right hand. A little reflection will show 
how the vessel may be steered in any other direction. An artificial steel magnet, fitted for this pur¬ 
pose in a proper box, is called the mariner" 1 s compass , or sea compass , or simply the compass ; which instru¬ 
ment is too well known to need any particular description. The mariner’s compass, with the addition 
of sights, divided circles, &c. for observing azimuths and amplitudes of the heavenly bodies, is called 
the azimuth compass. 


Def. X. The deviation of the horizontal needle from the meridian, or the angle 
which it makes with the meridian, when freely suspended in a horizontal plane, is 
called the declination or variation of the needle. 

PROP. X. There is generally a small variation in the direction of the magnetic 
needle, which differs in degree at dilferent places and times. 


This is known by observing the different points of the compass at which the sun rises or sets, and 
comparing them with the true points of the sun’s rising or setting, according to astronomical tables. 
Thus, if the magnetic amplitude is 80° eastward of the north, and the true amplitude is 82° toward the 
same side, then the variation of the needle is 2° west. The variation may be estimated from the 
azimuths in the same way. 

Schol. 1. A needle is continually changing the line of its direction, traversing slowly to certain lim¬ 
its toward the east and west. The first good observations on the variations were made by Burrowes 
about the year 1580, when the variation, at London, was 11° 15' east, and since that time the needle 
has been moving to the westward at that place ; also by the observations of different persons it has 
been found to point, at different times, as in the following table. 


Years. 

Observers. 

Variation E. or W. 

O ' 

Years. 

Observers. 

Variation E. or W. 
o / 

1580 

Burrowes. 

ii 

15 East. 

1723 

Graham. 

14 

17 West. 

1622 

Gunter. 

5 

56 

1747 


17 

40 

1634 

Gellibrand. 

4 

3 

1774 

Royal Soc. 

21 

16 

1640 

Bond. 

3 

7 

1775 

Royal Soc. 

21 

43 

1657 

Bond. 

0 

0 

1776 

Royal Soc. 

21 

47 

1665 

Bond. 

1 

23 West. 

1777 

Royal Soc. 

22 

12 

1666 

Bond. 

1 

36 

1778 

Royal Soc. 

22 

20 

1672 



2 

30 

1779 

Royal Soc. 

22 

28 

1683 



4 

30 

1780 

Royal Soc. 

22 

41 

1692 



6 

00 





By this table it appears, that from the first observation in 1580 till 1657, the change in the variation 

at London was 11° 15' in 77 yearS 

, which at a mean rate, is ne 

arly 9' a year. And 

from 1657 to 1780, 

it changed 22° 41', which is at the rate of 11' a year 

n » 

nearly. 


o 




f 1550 

8 

0 East. 


r 1600 

8 

0 East. 



1640 

3 

0 

At St. Helena the 1 . 1623 

6 

0 

At rans the Va- 

. 1660 

0 

0 

Variation 

of the) 111 1677 

0 

40 

IT cl XI 011 Ol 

tne < 

in 1681 

2 

2 West. 

Needle was ( 1692 

" 1 

0 West. 

Needle was 


1759 

18 

10 







L 1760 

18 

20 






Near the equator, in long. 40° east, the highest variation from the year 1700 to 1756, was 17° 15' 
west; and the least 16° 30' W. In lat. 15° N. and long. 60° W. the variation was constantly 5° E. In 
lat. 10° S. and long. 60° E. the variation decreased from 17° W. to 7° 15' W. In lat. 10° S. and long. 
5° W. it increased from 2° 15' to 12° 45' W. In lat. 15° N. and long, 20° W. it increased from 1° W. to 









86 


OF MAGNETISM. 


Book IV. 


9° W. In the Indian seas the irregularities were greater, for in 1700, the west variation seems to have 
decreased regularly from long. 50° E. to long. 100° E; but in 1756 the variation decreased so fast, that 
there was east variation in long. 80°, 85°, and 90° E. and yet, in long. 95° and 100° E. there was west 
variation. 

In the year 1775, in lat. 58° 17' S. and long. 348° 16' E. it was 0° 16' W. In lat. 2° 24' N. and long. 
32° 12' W. it was 0° 14' .45" W. In lat. 50° 6' 30'' N. and long. 4° O' W. it was 19° 28' W. 

Schol. 2. The variation of the needle is affected by heat and cold. The following is the result of 
observations made by Mr. Canton at different hours of the day, and also the mean variation for each 
month in the year. 


The Declination observed at different hours of the same 


Degrees of the 
Thermometer. 
62° 

62 

65 

67 

69 


Morning: 


Afternoon < 




day. 



June 27, 1759. 


Hours. 

Min. 

Declin. West. 

' 0 

18 

19° 

2' 

6 

4 

18 

58 

8 

30 

18 

55 

9 

2 

18 

54 

10 

20 

18 

57 

. 11 

40 

19 

4 

f 0 

50 

19 

9 

1 

38 

19 

8 

3 

10 

19 

8 

1 7 

20 

18 

59 

9 

12 

19 

6 

- 11 

40 

18 

51 


68i 


70 

70 

68 

61 

59 


The mean variation for each month in 
Year. 


i 


January - 

February 

March 

April 

May 

June - 

July 

August 

I ^ 

September 

October 

November 

December 


7 

•- 8 

11 

- 12 
13 

- 13 
13 

- 12 
11 

- 10 

8 

- 6 


the 


8 

58 

17 

26 

0 

21 

14 

19 

43 

36 

9 

58 


PROP. XI. A needle which, before it receives the magnetic power, rests on its 
centre parallel to the horizon, on becoming magnetical will incline toward the earth ; 
this is called the inclination or dip of the needle . 


Exp. Let a small dipping needle be carried from one end of a magnetic bar to the other; when it 
stands over the south pole, the north end ol the needle will be directed perpendicularly to it; as the 
needle is moved, the dip will grow less, and when it comes to the magnetic centre it will be parallel 
to the bar; afterward the south end will dip, and the needle will stand perpendicular to the bar when 
it is directly over the north pole. 

Schol. 1. This property of the magnetic needle was first discovered accidentally by Robert Norman, 
a compass-maker at Radcliffe, about the year 1576. He relates that it being his custom to finish and 
hang up the needles of his compasses, before he touched them, he found that immediately after the 
touch, the north point would always dip or incline downward, pointing in a direction under the horizon; 
so that, to balance the needle again he was forced to put a piece of wax on the south end as a counter¬ 
poise. The constancy of the effect led him to measure the angle which the needle would make with 
the horizon, and he found it at London to be 71° 50'. 

It is not yet absolutely ascertained whether the dip varies at the same place ; it is now, and has 
been since the year 1772, about 72°, according to several observations made by Mr. Nairne and the 
Royal Society. The trifling difference between the first observations of Mr. Norman, and these last of 
Mr. Nairne, &c. leads us to suppose that the dip is unalterable at the same place. 

It is certain, however, that the dip is different in different latitudes, and that it increases in going 
northward. It appears from a table of observations made with a marine dipping needle of Mr. Nairne, 
in a voyage toward the north pole in 1773, that 

In latitude 60° 18', the dip was 75° O'. 

In latitude 70 45, the dip was 77 52. 

In latitude 80 12, the dip was 81 52. 

In latitude 80 27, the dip was 82 2-^. 

See Phil. Trans. Vol. lxv. 


Schol. 2. The phenomena of the compass, and the dipping needle, and of the magnetism acquired 
by an iron bar in a vertical position, leave no room to doubt but that the cause exists in the earth. Dr. 
Halley supposed that the earth has within it a large magnetic globe, not fixed within to the external 
parts, having four magnetic poles, two fixed and two moveable, which will account for all the phenome- 


0 










Book IV. 


OF MAGNETISM. 


87 


na of the compass and dipping needle. This would make the variation subject to a constant law, where¬ 
as we find casual changes which cannot be accounted for upon this hypothesis. This the Doctor sup¬ 
poses may arise from an unequal and irregular distribution of the magnetical matter. The irregular 
distribution also of ferruginous matter in the shell may likewise cause some irregularities. 

Mr. Cavallo’s opinion is, that the magnetism of the earth arises from the magnetic substances there¬ 
in contained, and that the magnetic poles may be considered as the centres of the polarities of all the 
particular aggregates of the magnetic substances ; and as these substances are subject to change, the 
poles will change. Perhaps it may not be easy to conceive how these substances can have changed so 
materially, as to have caused so great a variation in the poles, the position of the compass having 
changed from the east toward the west about 33° in 200 years. Also the gradual, though not exactly 
regular change of variation, shows that it cannot depend upon the accidental changes which may take 
place in the matter of the earth. 

Mr. Churchman of America, says, there are two magnetic poles in the earth, one to the north and 
the other to the south, at different distances from the poles of the earth, and revolving in different 
times ; and from the combined influence of these two poles, he deduces rules for the position of the 
needle in all places of the earth, and at all times, past, present, or to come. 

The north magnetic pole, he says, makes a complete revolution in 426 years, 77 days, 9 hours, and 
the south pole in about 5459 years. In the beginning of the year 1777, the north magnetic pole was in 
76° 4' north latitude, and in longitude from Greenwich 140° cast; and the south was in 72° south lati¬ 
tude, 140° east from Greenwich. 


BOOK V 





OF ELECTRICITY. 


Def. I. The earth, and all bodies with which we are acquainted, are supposed 
to contain a certain quantity of ail exceedingly elastic fluid, which is called the electric 
fluid. 


Schol. This certain quantity belonging- to all bodies, may be called their natural share; and so long 
as each body contains neither more nor less than this quantity, it seems to lie dormant, and to produce 
no effect. 


Def. II. When any body becomes possessed of more or less than its natural quan¬ 
tity, it is said to be electrified , and is capable of exhibiting appearances which are as¬ 
cribed to the power of electricity. 

Schol. This equilibrium could never be disturbed, or, if it was disturbed, would be immediately re¬ 
stored, and therefore be insensible ; but that some bodies do not admit the passage of the electric fluid 
through their pores, and along their surfaces, though others do. 

Def. III. When a body has acquired an additional quantity of electric matter; or 
lost a part of what naturally belonged to it, and it is at the same time surrounded by 
bodies through which it cannot pass, it must remain in that state, and is said to be 
insulated. 


PROPOSITION I. 

The Electric Fluid, being excited, becomes perceptible to the senses. 

Exp. 1 . Let a long glass tube be rubbed with the hand, or with a leathern cushion; the electric fluid 
being thus excited, will attract light substances, and give a lucid spark to the finger, or any metallic 
substance, brought near it. 

The glass tube is called the electric , and all those bodies which are capable, by any means, of pro¬ 
ducing such effects, are called electrics. The hand, or any other body that rubs an electric, is called the 
rubber. 

2. As the exciting of a tube is very laborious for the operator, and the electricity procured by that 
means is small in quantity; globes and cylinders are used for this purpose. These, by a proper appara¬ 
tus, are made to revolve on their axes, and a rubber of leather is applied to the equatorial parts of the 
revolving glass, which become electrical by the friction. The electricity of the globe, or cylinder, is 
received by a metallic conductor insulated on a glass supporter. 

Plate 13, A cylinder or globe thus fitted up is called an electrical machine. C represents a glass cylinder 
Fig. 9. about 1 foot in diameter and 20 inches long, which is turned by means of a wheel ; the rubber or 

cushion is supported behind the cylinder by two upright springs that appear beneath, and are fastened 
to two cross bars of glass. B is a metallic conductor, supported on two pillars of glass ; from the end 
nearest to the cylinder issue several points, and at the other end the ball E projects by means of a wire. 
Sparks given by the conductor of a machine of this construction and magnitude are from 12 to 14 inches 
long. A chain D must connect the rubber with the earth. 



Book V. 


OF ELECTRICITY. 


89 


Schol. 1. In all experiments in electricity the greatest care should be taken to keep every part of 
the apparatus clean, and as free as possible from dust and moisture. When the weather is clear, and 
the air dry, especially in clear frosty weather, the electrical machine will always work well. But in 
very hot, or damp weather, the machine is not so powerful. 

Betore the machine is used, the cylinder should be first wiped very clean with a soft linen cloth ; 
and afterward with a clean hot flannel, or old silk handkerchief. 

Sometimes it will be necessary to apply to the rubber a very small quantity of amalgam made with 
one part of zinc, and four or five of mercury. 

Schol. 2. Respecting the theory of electricity, there are two different hypotheses, one that there is 
only one fluid, and the other that there are two. -Dr. Franklin’s hypothesis is the former, and it de¬ 
pends on the following principles. (1.) That all terrestrial bodies are full of the electric fluid. (2.) 
That the electric fluid violently repels itself, and attracts all other matter. (3.) By exciting an electric 
the equilibrium of the electric fluid contained in it is destroyed, and one part contains more than its 
natural quantity, and the other less. (4.) Conducting bodies, connected with that part which contains 
more electric fluid than its natural quantity, receive it, and are charged with more than their natural 
quantity ; this is called positive electricity ; if they be connected with that part which has less than its 
natural quantity, they part with some of their own, and contain less than their natural quantity; this is 
called negative electricity. (5.) When one body positively and another negatively electrified are con¬ 
nected by any conducting substance, the fluid in the body which is positively electrified rushes to that 
which is negatively electrified, and the equilibrium is restored. These are the principles of positive 
and negative electricity. The other hypothesis is, that there are'two distinct fluids, which was suggest¬ 
ed by M. Du Faye, upon his discovery of the different properties of excited glass, and excited resins, 
sealing-wax, &c. The following are the principles of this theory. (1.) That the two powers arise from 
two different fluids which exist together in all bodies. (2.) That these fluids are separated in non-elec¬ 
trics, by the excitation of electrics, and from thence they become evident to the senses, they destroying 
each other’s effects when united. (3.) When separated they rush together again with great violence, in 
consequence of their strong mutual attraction, as soon as they are connected by any conducting substance. 
These are the principles of vitreous and resinous electricity. 

PROP. II. The electric fluid passes easily along the surfaces of some bodies ; 
whilst other bodies do not convey it 5 the former are called Conductors - the latter Non¬ 
conductors, or Electrics. 

Exp. The metallic cylinder being fixed upon glass supporters, and placed near the electric machine, 
will, by means of the pointed wires, receive the electric fluid from the glass cylinder, and the fluid will 
be diffused over the whole surface of the metallic cylinder, from whence it cannot pass through the 
glass supporters which are electric, but may be conveyed away by any metallic or other conducting 
substances, brought near, or into contact with it. This metallic cylinder is called the Prime Conductor , 
or the Conductor. 

PROP. III. Some conductors are more perfect than others; and the electric fluid 
passes through that which is most perfect. 

Exp. The fluid will pass through a wire held in the hand. 

Schol. 1. The following bodies are conductors and electrics, disposed in the order of their degrees 
of perfection. Conductors; gold, silver, copper, brass, iron, tin, quicksilver, lead, the semi-metals, 
ores, charcoals, water, ice, snow, salts, soft stones, smoke, steam. Non-conductors, or Electrics; glass, 
and all vitrifications, even those of metals ; precious stones, resins, gums, amber, sulphur, baked wood, 
bituminous substances, wax, silk, cotton, feathers, wool, hair, paper, air, oil, hard stones. Many elec¬ 
trics become conductors, when heated, and all when moistened. 

Schol. 2. Glass vessels, made for electrical purposes, are often rendered very good electrics by use 
and time, though they might be very bad ones vyhen new. And some glass vessels, which had been long used 
for excitation, have sometimes lost their power almost entirely. Dr. Priestley mentions several instan¬ 
ces of very long tubes, which, when first made, answered the purposes of electricity admirably, but after 
a few months they have become almost useless. t 

Schol. 3. An exhausted glass vessel on being rubbed shows no signs of electricity upon its external 
surface. But the electric power of a glass cylinder is the strongest w hen the air within is a little rare¬ 
fied. If the air be condensed, or the cylinder be filled with some conducting substance, it is incapable of 
being excited. Nevertheless, a solid stick of glass, sealing-w r ax, sulphur, &c. may be excited. 

12 


90 


OF ELECTRICITY. 


Book V. 


Schol. 4. The same substance, by different preparations, is sometimes a conductor, and at others an 
electric. A piece of wood just cut from a tree is a good conductor;—let it be baked, and it becomes 
an electric ; burn it to charcoal, and it is a good conductor again lastly, let this coal be reduced to 
ashes, and these will be impervious to electricity. Such changes are also observable in many other 
bodies; and very likely in all substances there is a gradation from the best conductors to the best non¬ 
conductors of electricity. 

PROP. IV. Non-conductors retain the fluid on a small part of their surface where 
the friction has acted; conductors diffuse it over all their surface, and therefore can¬ 
not confine it, unless they be surrounded entirely by non-conductors, or be insulated. 

Exp. Observe the partial distribution of the lluid on an excited electric, and its universal diffusion 
over a conductor. If a linger, or any other conductor, be presented to an excited glass, cylinder, tube, 
&c. it will receive a spark, and in that spark, a small part only of the electricity of the electric; be¬ 
cause the excited electric being a non-conductor, cannot convey the electricity of all its surface to that 
point to which the conductor has been presented. But if any conducting substance be brought to a 
charged metallic conductor, it will receive in one spark nearly the whole of the electricity accumulated 
upon it. The small part which remains is very trifling in comparison of the first spark, and is called 
the residuum. 

Def. IV. A body is said to be positively electrified \ when it lias thrown upon it a 
greater quantity of the electric fluid than its natural share. 

Def. V. A body is said to be negatively electrified , when it has a less quantity of 
the electric matter than is natural to it. 

PROP. V. The electric fluid may he excited by rubbing, by pouring a melted elec¬ 
tric into another substanco, by heating and cooling, and by evaporation. 

Exp. 1. In working the electrical machine, the tluid is excited by friction. Rubbing is the general 
mean by which all electric substances that are at all excitable may be excited. Whether they be rub¬ 
bed with electrics of a different sort, or conductors, they ahvays show signs of electricity, and in general 
stronger when rubbed with conductors, and weaker when rubbed with electrics. 

2. When sulphur is melted into an earthen vessel, if the vessel be supported by a conducting sub¬ 
stance, the sulphur, when cold and separated from the vessel, is strongly electrical, and will attract 
light bodies. 

3. If sulphur be melted into glass vessels,’when cold, the glass, whether supported by electrics or 
not, will be positively electrified, and the sulphur negatively. 

4. Melted sealing-w r ax, when poured into sulphur, becomes positively electrified, and the sulphur 
negatively. 

5. Melted sealing-wax poured into glass cups acquires a negative electricity; upon being separated, 
the glass is positive. 

6. Sulphur, melted into metallic cups, shows no signs of electricity till it is separated from the cup, 
when the cup is negative and the sulphur is positive. 

7. if a stick of sealing-wax be broken into two pieces, the extremities that were contiguous will be 
found electrified, one positively, and the other negatively. 

8. The tourmalin, a stone which is generally of a deep red, or purple colour, about the size of a 
walnut, and found in the East Indies, while kept in the same degree of heat, shows no signs of electrici¬ 
ty, but will become electrical by increasing or diminishing its heat, and stronger in the latter than in 
the former case. (1.) Its electricity does not appear all over its surface, but only on tw o opposite sides, 
which may be called its poles, and they are always in one right line with the centre of the stone, and 
in the direction of the strata ; in which direction the stone is absolutely opaque, though on the other 
side it is semitransparent. (2.) Whilst the tourmalin is heating, one of its sides (call it A) is electrified 
plus; the other (call it B) minus. But when it is cooling, A is minus, and B is plus. (3.) If this stone 
be excited by friction, then both its sides at once maybe made positive. (4.) If a tourmalin be cut into 
several parts, each piece will have its positive and negative poles, corresponding to the positive and 
negative sides of the stone from which it was cut. 

Schol. These properties are now found to belong to several hard and precious stones, as well as 
to the tourmalin. 

9. Electricity may be produced by the evaporation of water in this manner:—Upon an insulat-' 


Book V. 


OF ELECTRICITY. 


91 


ing stand, as a wine glass, place an earthen vessel, as a crucible, a basin, &c. and put into it three or 
four lighted coals. Let a wire be put with one end among the coals, and with the other let it touch a 
very sensible electrometer. Then pour in a spoonful of water at once upon the coals, which will occasion 
a quick evaporation; and at the same time the electrometer will diverge. For a description of the 
electrometer, see Prop. XII. Schol. 

PROP. VI. The electric fluid may be lodged in electrics, or in insulated conduc¬ 
tors, in a greater quantity than naturally belongs to them, or they may be positively 
electrified. 

Exp. In Working the machine, the cylinder acquires more than its natural quantity of fluid by excit¬ 
ation, the conductor, by communication; for, while there is a free conveyance of fluid from the earth 
to the rubber, by means of a conducting supporter, the conductor will be highly electrified. 

Schol. The electric matter with which the prime conductor is loaded, is not produced by the friction 
of the cylinder against the rubber. It is only collected by that operation from the rubber, and all the 
bodies that are contiguous to it. If, therefore, the rubber be well insulated, the friction of the cylinder 
will produce but little electricity ; for in that case the rubber can only part with its own share, which 
is very inconsiderable. In this situation, if the finger be presented to the rubber, sparks will be seen 
to dart from it to the rubber, to supply the place of that electric matter which had passed from it to the 
cylinder; if the conductor be also insulated, these sparks will cease as soon as it is fully loaded. 

PROP. VII. The electric fluid being accumulated on any body will pass to any 
conductor brought near to the body ; if it pass from, or be received by, pointed wires, 
it will be conveyed in a continued stream ; if it pass from, or be received by, a surface 
which has no sharp points, it will be discharged with an instantaneous explosion or spark. 

Exp. 1 . Receive the fluid from the conductor upon a pointed wire, and upon a brass ball. 

2. The fluid will be diffused thi’ough the surrounding atmosphere, by wires placed upon the con¬ 
ductor. 

Cor. Hence arises the necessity of keeping the whole surface of the conductor free from points. 

Schol. When a conductor is electrified by communication, its whole electric power is discharged atf 
once, on the near approach of a conductor communicating with the earth ; whereas an excited electric, 
in the same circumstances, loses its electric power only in the parts near to the conductor. 

PROP. VIII. If conductors be insulated, they will retain a greater or less quan¬ 
tity of the electric fluid (the power of the machine being given) proportional to the 
extent of surface in the conductor. 

Exp. Observe the difference in the magnitude and distance of sparks taken from a small conductor, 
and of those taken from a large one. 

There is a limit, beyond which this proposition will not hold true, but which experiment has not 
yet ascertained. For it is certain, that if the conductor be very long, it will discharge itself over the 
cylinder back to the rubber long before it is fully charged. The late Mr. G. C. Morgan, whose memory 
will be ever dear to the editor of this work, asserts, that by the most powerful excitation of a cylinder, 
14 inches in diameter, the spark afforded by a conductor 8 inches in diameter, and 12 feet long, did not 
equal half the length of that procured from the same cylinder with a conductor of equal diameter, but 
shortened to 6 feet. And he thinks that a conductor of half that length even, and about 16 inches in 
diameter, would have yielded a longer spark than either of the preceding. See Morgan’s Lect. on 
Elect. Vol. I. p. 54, &c. * 

PROP. IX. A body may be deprived of part of its natural portion of electric 
fluid, or be negatively electrified. 

Exp. If the rubber which communicates the fluid to the glass cylinder, and from thence to the con¬ 
ductor, be insulated, because by working the machine a quantity of its fluid is conveyed away, and it 
cannot receive a fresh supply through its supporter, it will be in an exhausted or negative state. 

Schol. If negative electricity be required, then the chain which connects the rubber with sur¬ 
rounding objects, and consequently with the earth, the great reservoir of the electric fluid, must be re¬ 
moved from the insulated rubber, and hung to the prime conductor; for in this case the electricity of 
the conductor will be communicated to the ground, and the rubber will appear strongly negative. 
Another conductor may be connected with the insulated rubber, and then as strong negative electricity 
may be obtained from this as positive can be in the case before mentioned. 


92 


OF ELECTRICITY. Book V. 

The patent machine of Mr. Nairne is admirably adapted for the purposes both of positive and nega¬ 
tive electricity. 

PROP. X. When bodies are negatively electrified, they receive the fluid from other 
bodies brought near them. 

Exr. 1. Let two insulated conductors, one of which is connected with the glass cylinder, the other 
with the rubber, be electrified ; whilst they are in this state let them be brought near each other ; a 
spark will pass from that which (by Prop. VI.) is positively, to that which (by Prop. IX.) is negatively 
electrified. 

2. Let two persons standing on glass feet be electrified, first, both positively, or both negatively, 
they will not, on contact, communicate the fluid to each other; but let them be electrified, the one posi¬ 
tively and the other negatively, by making a communication from one to the conductor, and from the 
other to the rubber; on contact, the former will give, and the latter receive a spark. 

PROP. XI. From a pointed body positively electrified the fluid will be seen to 
stream out, toward any uneleetrified body brought near it, in a conical pencil of rays ; 
whereas, in passing from the unelectrified body to a pointed body negatively electrified, 
it will form a globular flame, or star, about its point. 

Exp. 1. Observe in a dark room, the different appearances of the electric fluid at the extremity of a 
pointed wire, when the point is presented to an insulated conductor positively, and when it is presented 
to one negatively, electrified ; or when such a wire is fixed upon a conductor positively or negatively 
.electrified. 

2. Within a luminous conductor electrified positively, (viewed in a dark room) the fluid will be seen 
passing in the form of a peucil from one wire, and received in the form of a star upon the other; and 
the reverse if it be electrified negatively. 

PROP. XII. If two bodies be electrified, both positively, or both negatively, they 
repel each other ; but if one be electrified positively, and the other be negatively or not 
at all electrified, they attract each other. 

Exp. 1. Light feathers, or hair, connected with the conductor, appear repellent, but are attracted by 
bringing any non-electrified body near them. 

2. The hair of a person electrified becomes repellent. 

3. In the graduated electrometer the ball is repelled according to the degree in which the conductor 
is electrified. 

4. Downy feathers, paper figures, threads of flax, thistle down, gold leaf, brass dust, or other light 
bodies, brought near to the conductor, are altei'nately attracted and repelled. This will not take place 
if the bodies be laid on a plate of glass. 

5. Two bells being suspended by wires from a brass rod connected with the conductor, and a third 
by a silk cord, and two small balls of brass suspended by a silken thread between the bells, the fluid will 
be communicated from the conductor to the outer bells, and by the balls to the middle bell, and from 
thence conveyed by a chain to the earth ; the balls in receiving and communicating the fluid are attract¬ 
ed and repelled succesively, and produce ringing. 

6. Let water flow from a capillary tube, from which, before it is electrified, it passes in drops; up¬ 
on being electrified, the particles of fluid will be separated, and their motion accelerated. 

These appearances will be presented, whether the conductor be positively or negatively electrified. 

7. Mr. Symmer, in the year 1759, presented to the Royal Society some papers upon the electricity 
of silk stockings. He had been accustomed to wear two pairs of silk stockings, a white pair under 
black. When these were pulled off together, no signs of electricity appeared, but on pulling off the 
black from the white, he heard a snapping noise, and in the dark perceived sparks of fire. On this 
subject he has related a number of very curious experiments on the attraction and repulsion of the 
stockings, and upon their different states of electricity. 

Cor. Since it is found that rubbed glass electrifies any insulated conductor positively, it may be de¬ 
termined whether any body is electrified positively or negatively, by bringing it near to a pith-ball, or 
down-feather, positively electrified, and observing whether the ball or feather be attracted or repelled 
by the body. 

Exp. Bring a pith-ball or down-feather, suspended by a silken thread and positively electrified by 
any rubbed glass surface, near to another pith-ball or feather suspended by a flaxen thread from a con¬ 
ductor connected with the cylinder ; then bring the same near to a conductor connected with the 
rubber. 


Book V. 


OF ELECTRICITY. 


93 


Schol. The electrometer is an instrument invented to measure the degree of electrification of any Plate 13. 
body. Small degrees of electricity are shown by the divergence of two very small pith-balls, a, 6, sus- F, S- im¬ 
pended upon parallel threads, straws, &c. These balls presented to a body in its natural state will not 
be affected; but if the body be electrified, they will be attracted by it and diverge. 

Another very useful and common electrometer consists of an upright stick, AB, to which is affixed a Plate 13 
graduated semicircle ; D is a pith-ball stuck upon the end of a fine straw, which by means of an axis at Fig. D. 
C, is moveable in a plane parallel to that of the semicircle. This electrometer is fixed upright on a 
prime conductor; and when it is not electrified, the radius will hang down, and according to the intensi¬ 
ty of the electric state given to the conductor, the repulsion must cause the ball to ascend. The ascent 
will be marked by the graduations. 

Mr. Cavallo has invented a very sensible electrometer, well adapted for the observation of the Plate'13. 
presence and quality of natural and artificial electricity. ABC is the brass case containing the instru- F *g- 12 
ment. When the part AB is unscrewed, and the electrometer taken out, it appears as represented in 
ABDC. A glass tube, CDNM, is cemented into the piece AB. The upper part of the tube is shaped 
tapering to a small extremity, which is entirely covered with sealing-wax. Into this tapering part a 
small tube of glass is cemented; the lower extremity being also covered with sealing-wax projects a 
small way within the tube CDNM. Into this smaller tube, a wire is cemented, which, with its under 
extremity, touches a flat piece of ivory H, fastened to the tube by means of a cork. The upper ex¬ 
tremity of the wire projects about a quarter of an inch above the tube, and screws into the brass cap 
EF, which cap is open at the bottom, and serves to defend the waxed part of the instrument from the 
rain. From H are hung two fine silver wires, having very small corks at the lower ends, which, by 
their repulsion, show- the electricity. IM, and KN, are two slips of tin-foil stuck to the inside of the 
glass, and communicating with the brass bottom AB. They serve to convey away that electricity, 
which when the corks touch the glass, is communicated to it, and might disturb their free motion. 

When this instrument is used to observe artificial electricity, it is set on a table, and electrified by 
touching the brass cap EF with an electrified body ; in this state, if any electrified substance is brought 
near the cap, the corks of the electrometer, by their converging, or diverging more, will show the 
species of electricity. 

When it is to be used to try the electricity of fogs, &c. it must be unscrewed from its case, and held 
a little above the head by the bottom AB, so that the observer may conveniently see the corks, which 
will immediately diverge if there is any sufficient quantity of electricity in the air, the nature of which 
may be ascertained by bringing an excited piece of sealing-wax toward the brass cap EF. 

PROP. XIII. From the sharp points of electrified bodies there proceeds a current 
of air. 

Exp. 1. A wire with sharp points bended in opposite directions, and suspended on the point of a 
perpendicular wire inserted in the conductor, will be carried round by the current proceeding from 
the points. 

2. Let several pieces of gilt paper be stuck like vanes into the sides of a cork, through the centre 
of which a needle passes ; suspend the whole by a magnet, and present one of the vanes to the point 
of a wire inserted in the conductor; they will be put into motion. 

PROP. XIV. Some bodies, upon being rubbed, are electrified positively, and 
others negatively; and the same bodies are capable of being electrified positively, or 
negatively, as they are rubbed with different substances. 

Exp. Smooth glass becomes positively electrified by being rubbed with any substance hitherto tried, 
except the back of a living cat; rough glass becomes positively electrified by being rubbed with dry 
oiled silk, sulphur, and metals; negatively, with woollen cloth, sealing-wax, paper, the human hand. 

White silk becomes positively electrified by being rubbed with black silk, metals, black cloth ; negative¬ 
ly, with paper, hairs, the hand. Black silk will be positively electrified with red sealing-wax; 
negatively, with hare’s skin, metals, the hand. Sealing-wax will be negatively electrified with the 
hand, leather, woollen cloth, paper, hare’s skin. Baked wood will be positively electrified with silk; 
negatively with flannel. If these and other substances, being electrified, be brought near to a pith- 
bail or down-feather, as described Prop. XII. Cor. Exp. it will appear whether they are electrified 
positively or negatively. 

PROP. XV. Bodies insulated, if placed within the influence of an electrified body, 
will be electrified, at the part adjacent to that body, in the manner contrary to that 
of the electrified body. 


94 


OF ELECTRICITY. 


Book V. 


Plate 13. 
Fig. 13. 


Exp. 1 . Bring a conductor (without pointed wires) near to the glass cylinder, whilst the machine 
is working; if the conductor be not insulated, it will be negatively electrified till it is brought so near 
as to receive sparks from the cylinder; if the conductor be insulated, it will, in the same situation, 
be electrified negatively, in the parts nearest the cylinder, and positively in the parts more remote; 
as may be seen by bringing an excited glass tube (which is positively electrified) near to a ball sus¬ 
pended from the conductor. Compare Prop. XII. Cor. 

2. Let two pith-balls be so suspended by flaxen threads as to be in contact w r hen unelectrified ; 
on being brought near to a body electrified positively, they will repel each other, being electrified 
negatively : if the balls be suspended in the same manner by silken threads, they will, in the same 
situation, be positively electrified. 

3. Let PC be an electrified prime conductor, and AB a metallic body placed within its atmosphere, 
but beyond the striking distance. Now from the principles already explained, it is evident that the 
electrical atmosphere of the prime conductor must be positive or negative. (1.) If it be positive, then 
the adjacent part A of the metallic body AB, will be found to be electrified negatively; the remote 
part B, will be electrified positively ; and there will be a certain point I), in its natural state, or not elec¬ 
trified at all. (2.) If the prime conductor be charged with negative electricitjq then A will be positive, 
B, negative, and still some point, as D, w ill be found unelectrified, which is called the neutral point. 

Earl Stanhope has demonstrated, by a considerable number of experiments, that the neutral point 
D is the fourth point of a harmonical division of the line CAB. Consequently the points C, A, and B 
being given, the neutral point D may be always found. For by the proportion assumed by his lordship, 
as the whole line BC is to the part CA, so is the remote part BD to the middle term DA; therefore 
by composition, 

BC -f CA (BA -F 2AC) : CA: : BD + DA (BA) : AD. 

Thus, if BA be 40 inches, and CA 36, then AD is equal to 12-f- inches. 

Cor. 1. From the nature of this proposition, it is evident that the neutral point D can never be 
farther from A than half the distance between A and B, supposing the electrified conductor PC to be 
removed to an infinite distance. 

Cor. 2. It is likewise evident, that the evanescent position of the neutral point D must be A, when 
the end A of the metallic body AB comes into contact with the charged body PC. 

Schol. From the above considerations, Lord Stanhope has, with great ingenuity, proved by an 
elaborate mathematical demonstration, illustrated and confirmed by a great variety of experiments, that 
the density of an electrical atmosphere superinduced upon any body must be inversely as the square 
of the distance from the charged body. 

4. Let a circular plate composed of resin and sulphur, or of sealing-wax, be negatively electrified 
by rubbing it with flannel; whilst it is in this state, let a metallic plate of the same form and size, hav¬ 
ing a glass handle fastened to its centre, be placed, by means of the handle, on the electrified plate ; 
then receive a spark from the metallic plate with the finger; after which the metallic plate, being 
removed by the glass handle, will be found to be positively electrified. This instrument is called an 
electrophorus. 

5. Let one side of a plate of glass be electrified positively, the other side will attract light bodies, 
being negatively electrified. 

6. Let a plate of glass be placed between two metallic plates about two inches in diameter small¬ 
er than the plate of glass, and let the plates be supported by a conductor; upon positively electrifying 
the upper metallic plate, by means of a wire connected with the prime conductor, the fluid not 
being able to pass along the glass, will be accumulated upon the part contiguous to the upper metallic 
plate; w'hilsl the low'er metallic plate, being within the electric influence of the upper, will be 
negatively electrified. 

PROP. XVI. When any electric substance is electrified, it will continue in that 
state till some conductor conveys away the accumulated or restores the deficient 
fluid ; which will be done more or less rapidly, according to the degree of conducting 
power in the conductor, and the number of points in which it touches the electric. 

Exp. 1 . When the metallic plate in the electrophorus is electrified (as described Prop. XV. Exp. 4.) 
by setting it upon the electric plate, touching it with the finger, and separating it successively, many 
sparks may be obtained, w ithout again exciting the electric plate; for this plate being negatively 
electrified, the metallic plate on being touched with the hand, becomes positively electrified (by Prop. 
XV.) and tbe electric plate remains long in its negative state, because not being a conductor, its defi¬ 
ciency will be slowly supplied from the air where its surface is not covered. 

2. If a glass vessel, a common drinking-glass, for instance, held in the hand, receive the electric 
fluid on the inside from a wire, or chain, fixed on the conductor, pith-balls, placed under the vessel 
upon a conducting supporter, will continue long in motion. 


Book V. 


OF ELECTRICITY. 


95 


3. Let a plate of glass be electrified in Hie manner described in Prop. XV. Exp. 6. Because one 
side of the plate is positively electrified, and the other negatively, if a communication be made from 
one metallic plate to the other by means of some conductor, part of the accumulated fluid will suddenly 
pass to the side which is deficient; upon a second application of the plates of metal to the glass, there 
will be a second explosion. 

Schol. AB is an electric jar, coated with tin-foil on the inside and outside, within three inches of 
the top, having a wire with a round bi'ass knob K, at its extremity. This wire passes through the cork * lg ' 

D, that stops the mouth of the jar, and, at its lower end, is bended or branched so as to touch the inside 
coating in several places. Coated jars may be made of any form and size, and are called Leyden Phials , 
or Leyden Jars . 

A number of jars combined, make what is termed an electrical battery they all stand in a box, the 
the bottom of which is covered with tin, thus all their outsides are connected ; and by means of wires 
and brass rods, their insides are also connected. 

The discharging rod consists of a glass handle A, and two curved wires BB, which move by a joint P[ ate 
C, fixed to the brass cap of the glass handle A. The wires BB are pointed, and the points enter the * l °' 5 * 
knobs DD, to which they are screwed, and may be unscrewed from them at pleasure. By this con¬ 
struction, the balls or points may be used as occasion requires. The wires being moveable at the joint 
C, may be adapted to smaller or larger jars at pleasure. 

PROP. XVII. If a glass plane, or cylindrical vessel, coated on both sides with tin- 
foil, or any other conducting substance, be charged , that is, positively electrified on 
one side, and consequently negatively electrified on the other; a communication being 
made from one side to the other by some conductor, the plane, or vessel, will he sud¬ 
denly discharged , with an explosion. 

There is a strong attraction (compare Prop. XII. and XV.) between the fluids on opposite sides of 
the glass, or the fluid which is accumulated on one side makes a powerful effort toward the other side 
where the fluid is deficient; but the substance of the glass itself being impervious to the electric fluid, 
the accumulated fluid cannot pass to the deficient side till a communication is made between them by 
some conducting substance. When such a communication is made, because the metallic coating touches 
the whole surface of the electrified glass, the whole quantity of redundant fluid easily passes from the 
side which was positively electrified to the other. 

Exp. 1 . Let a plate of glass coated with tin-foil (except about 1 ^ inch from the edge) be charged, as 
described in Prop. XV. Exp. 6. Upon making a communication from one side to the other by the discharg¬ 
ing rod, there will be a sudden discharge. 

2. Let the same be done with the Leyden Phial. 

3. Charge a jar, coafed on the inside, with water, shot, or brass dust, and held on the outside by the 
hand ; then discharge it in a dark room. 

4. If tw o equal circular brass plates, one of which is suspended by a long metallic rod from the con¬ 
ductor parallel to the horizon, and the other, supported by a conductor, is placed parallel and opposite 
to the first, be electrified ; the plate of air between them will be charged by the brass plates. 

5. Let one coated jar be suspended by a wire under another; let the upper jar be charged by taking 
sparks from the conductor ; the lower uninsulated jar will be charged with the fluid which passes from 
the side negatively electrified of the upper jar. 

6. Discharge, in a dark room, a jar imperfectly coated. 

Cor. 1 . A coated jar cannot be charged unless its outer surface be connected with some conductor. 

For without such a conductor, the fluid cannot pass from or to the outer surface, which is necessary in 
order to charge the jar. 

Cor. 2. When a coated glass vessel is charged, the charge of electric fluid is in the o-lass and not in 
the coating. 

Exp. Lay a plate of glass between tw r o metallic plates, as described Prop. XV. Exp. 6. Having 
charged the plate of glass, remove the upper plate of metal by a glass handle, with some non-conductin°- 
sublance, as silk; remove the electrified glass plate, and place it between two other plates„of metal un- 
electrified and insulated; the plate of glass thus coated afresh will still be charged. 

Schol. The discharge of a plate of glass, Leyden Phial, &c. is made by restoring the equilibrium 
which was destroyed by the charging; and it is effected by forming a communication°between the over¬ 
loaded and the exhausted side ; and if the communication be made by metal, or other o 0 od conductors 
the equilibrium will be restored with violence, the redundant electricity on one side wilfrushwith great 
rapidity through the metallic communication to the exhausted side, and a large explosion will be made, 
that is, the flash of electric light will be very visible, and the report will be loud. 


96 


OF ELECTRICITY. 


Book V. 


PROP. XVIII. If the conductor be electrified positively, that side of the jar with 
which it has a communication will be electrified positively, the other negatively, 

Exp. 1. Charge one jar on the inside positively, and another negatively, and observe, in a dark room 
the different appearances of the fluid, upon the point of a wire brought near to the ball which is con¬ 
nected with the inner side of each jar : when the point is presented to the jar positively electrified on 
the inner side, it will exhibit the appearance of a star; when presented to the other, that of a 
pencil. 

2. Observe the different appearances, in a dark room, when with the same charged jar the point is 
presented toward the side positively, and toward the side negatively, electrified. 

3. Between two jars, charged one negatively and the other positively, suspend by a silken string 
a cork ball, from which short threads hang freely; the ball will pass with a rapid motion from one to 
the other, and, being first attracted toward the jar positively electrified, then toward the other, it will 
receive the fluid from the former, and communicate it to the latter, till both are discharged. If both be 
charged in the same manner, the cork will remain at rest. 

4. If, after a jar is charged, the uncoated part of the jar be moistened by the breath, or by steam, 
the jar placed upon a conductor will be gradually discharged, and the fluid will be seen, in a dark room, 
to flash strongly from one side to the other; if the jar be insulated, the flashes will be greatest on the 
side positively electrified. 

5. Let a discharging rod be applied without its balls to a charged jar, in such a manner as to dis¬ 
charge the jar gradually; the point which approaches toward the side positively electrified, will, in a 
dark room, exhibit a star; the other point, a pencil. 

6. Within the receiver of an air-pump place two well polished brass balls, the lower supported on a 
brass stem by the plate of the pump, the other fixed on a stem which is moveable in the neck of the 
receiver ; let the balls be brought within the distance of four or five inches from one another; then let 
the upper ball be connected with the conductor, and electrified positively ; a lucid atmosphere will, in 
a dark room, appear on the lower surface of the upper ball; whereas if the upper ball be negatively 
electrified, the lucid atmosphere will be seen on the lower ball. 

PROP. XIX. The electric fluid can be conveyed through an insulated conductor 
of any length, and its passage from one side of a charged jar to the other, is apparent¬ 
ly instantaneous, through whatever length of a metallic, or other good conductor, it is 
conveyed. 

Exp. 1. Let a long wire, passing round a room, suspended by silk cords, be a part of the circuif of 
communication from one side of a charged jar to the other; the discharge will be apparently at the 
same instant in which the communication from one side to the other is completed. 

2. Let any number of persons make a part of the circuit of communication ; the fluid will pass in¬ 
stantaneously through the whole circuit. 

Schol. The shock of the Leyden jar has been transmitted through wires of several miles in length, 
without taking any sensible space of time. Dr. Priestley relates several curious experiments made with 
a view of ascertaining this point soon after the invention of the Leyden Phial. See Priestley’s Hist, of 
Elect. 

PROP. XX. The sudden discharge of a charged jar gives a painful sensation to 
any animal, placed in the circuit of communication, called the electric shock. 

The discovery of the effects of electricity, as exhibited by the Leyden jar, immediately drew the 
attention of all the philosophers in Europe. The account which some of them gave of the experiments 
to their friends, border very much on the ludicrous. M. Musschenbfoeck, who tried the experiment 
with a glass bowl, told M. Reaumur, in a letter written soon after the experiment, that he felt himself 
struck in his arms, shoulder, and breast, so that he lost his breath ; and it was two days before he re¬ 
covered from the effects of the blow and the terror. He added, that he would not take a second shock 
for the whole kingdom of France. 

M. Allamand, who made the experiment with a common beer glass, said, that he lost his breath for 
some moments, and then felt such an intense pain all along his right arm, that he was apprehensive of 
bad consequences ; but it soon went off without any inconvenience. 

Notwithstanding the parade made by these philosophers, the shock was probably, not by any means 
stronger than what many children 6 or 7 years old would bear without the smallest hesitation. Their 
descriptions must have arisen from terror, or love of the marvellous. 


Book V. 


OF ELECTRICITY. 


97 


Cor. The force of the electric shock may be increased, by increasing the surface of the coated 
glass. 

Exp. 1. A battery being charged, a fine metallic wire brought into the circuit will be melted. 

2. If a plane piece of metal be placed upon one of the rods of the discharger, and upon the other a 
needle with the point opposite to the surface of the metal, upon discharging the battery, the surface of 
the piece of metal will be marked with coloured circles, occasioned by thin laminse of metal raised in 
the explosion. 

3. If a piece of gold-leaf be put between two pieces of glass, and the whole fast bound together, the 
metal will be melted, and a metallic stain will be seen in both glasses. 

4. If a shock be sent through a needle, it will give it magnetic polarity. 

5. An animal or plant may be killed by being placed in the circuit of a battery. 

Schol. Persons, not thoroughly conversant in electricity, should be very cautious in using large 
batteries; they should be sure that they are perfect masters of a small force, before they meddle with a 
greater. Such a force of electricity as may be accumulated in batteries is not to be trifled with, since 
the consequences, if not fatal, may be great and lasting. A large shock, taken through the arms and 
breast, which'an operator is most in danger of receiving, might possibly injure the lungs, or some other 
vital part; and if the shock were taken through the head, which may easily happen when a person is 
stooping over the apparatus in order to adjust it, it might affect his intellects for the remainder of life. 

PROP. XXI. If the circuit be interrupted, the fluid will become visible, and where 
it passes, it will leave an impression upon any intermediate body. 

Exp. 1. Let the fluid pass through a chain, or through any metallic bodies placed at small distances 
from each other; the fluid, in a dark room, will be visible between the links of the chain, or between 
the metallic bodies. 

2. If the circuit be interrupted by several folds of paper, a perforation will be made through them, 
and each of the leaves will be protruded by the stroke from the middle toward the outward leaves. 

3. Let a card be placed under wires which form the circuit, w here the circuit is interrupted for the 
space of an inch ; the card will be discoloured. If one of the w ires be placed under the card, and the 
other above it, the direction of the fluid may be seen. 

4. Spirits of wine, or gunpowder, being made part of the circuit, may be fired. 

5. Inflammable air may be fired by an electric gun. 

PROP. XXII. The atmosphere is electrified, sometimes positively, and sometimes 
negatively. 

Exp. Let a kite be sent up into the air with cord, consisting of copper thread twisted with twine; 
let the lower end of the cord be insulated by a silk line ; a metallic conductor suspended from the lower 
end of the cord will be positively or negatively electrified. The air at some distance from houses, trees, 
masts of ships, &c. is generally electrified positively ; particularly in frosty, clear, or foggy weather. 
For the particular construction of the electrical kite, and other instruments used with it, see Cavallo’s 
Elect. Vol. ii. Chap. 1. 

Schol. The following general laws have been deduced by Mr. Cavallo, from a great number of ex¬ 
periments made during two years in almost every degree of the atmosphere from 15° to 80° of Fahren¬ 
heit’s thermometer. 

1. The air appears to be electrified at all times; its electricity is constantly positive, and much 
stronger in frosty than in warm weather; but it is by no means less in the night than in the day time. 

2. The presence of clouds generally lessens the electricity of the kite. 

3. When it rains, the electricity of the kite is generally negative, and very seldom positive. 

4. The aurora borealis seems not to affect the electricity of the kite. 

5. The electrical spark, taken from the string of the kite, or from any insulated conductor connect¬ 

ed with it, especially if it does not rain, is very seldom longer than 1 of an inch, but it is exceedingly 
pungent. When the index of the electrometer is not higher than 20°, the person that takes the spark 
will feel the effect of it in his legs ; it appearing more like the discharge of an electric jar, than the 
spark taken from a prime conductor. « 

6. The electricity of the kite is in general stronger or weaker, according as the string is longer or 
shorter; but it does not keep any exact proportion to it. The electricity, for instance, brought down 
by a string of an hundred yards, may raise the index of the electrometer to 20°, wdien with double that 
length of string the index of the electrometer will not go higher than 25°. 

13 


98 


OF ELECTRICITY. 


Book V. 


• 7. When the weather is damp, and the electricity is pretty strong 1 , the index of the electrometer, 
after taking a spark from the string, or presenting the knob of a coated phial to it, rises surprisingly 
quick to its usual place, but in dry and warm weather it rises exceedingly slow. 

PROP. XXIII. The electric fluid and lightning are the same substance. 

Their properties and effects are the same. Flashes of lightning are generally seen to form irregu¬ 
lar lines in the air ; the electric spark, when strong, has the same appearance. Lightning strikes the 
highest and most pointed objects; takes in its course the best conductors; sets lire to bodies; some 
times dissolves metals ; rends to pieces some bodies ; destroys animal life ; in all of which it agrees (as 
has been shown) with the phenomena of electric fluid. Both causes have the same power of making 
iron magnetic. Lightning has been known to strike men with blindness. Dr. Franklin produced a 
similar effect on a pigeon by the electrical fluid. Lastly, the lightning being brought from the clouds 
to an electrical apparatus, by a kite or wire, will exhibit all the appearances of the electric fluid. 

Exp. Take a Leyden phial, 5 inches in diameter, and 13 inches in height; on the inside let the 
coating rise till its upper edge be 2-^ inches from the rim of the vessel; on the outside let the coating 
rise no higher than one inch from the bottom. When the phial is thus coated, let it be charged, and a 
spark will pass from the tin-foil on the outside to that on the inside; but its form will resemble that of 
a tree, whose trunk will increase in magnitude and brilliancy, and consequently in power, as it ap¬ 
proaches the edge, owing to ramifications which it collects from all parts of the glass. Within two 
inches of the edge, it becomes one body, or stream, and along that interval its greatest force acts. 

When two clouds, or the two correspondent parts of a cloud, have their equilibrium restored by a 
discharge, th<? appearances are exactly similar to those of the preceding experiment. Each extremity 
of the flash is formed by a multitude of little streams, which gather into one body, whose power is un¬ 
divided in that interval only which separates the positive from the negative. 

PROP. XXIV. Buildings may be secured from the effects of lightning, by fixing a 
pointed iron rod higher than any part of the building, and continuing it, without inter¬ 
ruption, to the ground, or the nearest water. 

The electric fluid will, by means of the pointed rod, be gradually conveyed from the cloud to the 
earth by a continued stream, and thus prevent the effects of a sudden and violent explosion. 

Exr. Let a board, shaped like the gable end of a house, be fixed perpendicularly upon a horizontal 
board ; in the perpendicular board let a hole be made, about an inch square and ^ inch deep; in this 
hole let a piece of wood nearly of the same dimensions be so inserted as to fall easily out of its place, 
and let a wire be fastened diagonally to this square piece of wood ; let another wire, terminated by a 
brass ball, be fastened to the perpendicular board, with its ball above the board, and its lower end in 
contact with the diagonal wire in the square piece of wood ; let the communication be continued by a 
wire to the bottom of the perpendicular board. If the wires in this state be made part of a circuit 
of communication, on discharging the jar the square piece of wood will not be displaced ; but if the 
communication be interrupted by changing the direction of the diagonal wire, the square piece of 
wood will, upon the discharge, be driven out of its place 

If instead of the upper brass ball, a pointed wire be placed above the perpendicular board, the 
discharge may be drawn off without an explosion. 

Schol. The following directions are given by Earl Stanhope, to persons erecting conductors of 
lightning. 

(1.) The rods must be made of such substances as are, in their nature, the best conductors of elec¬ 
tricity. 

(2.) The rods must be uninterrupted, and perfectly continuous. 

(3.) They must be of sufficient* thickness. 

(4.) They must be perfect!}' connected with the common stock, that is, the earth, or the nearest 
water. 

(5.) The upper extremity of the rods must be finely tapered, and as accurately pointed as possible. 

(6.) The rods must be very prominent, and several feet above the chimneys. 

(7.) Each rod must, be carried in the shortest convenient direction from its upper end to the com¬ 
mon stock. 

(8.) There should be no.prominent bodies of metal on the top of the building proposed to be se¬ 
cured, but such as are connected with the conductor by some proper metallic communication. 

(9.) There should be a sufficient! number of substantially erected, high, and pointed rods. See 

* Perhaps ^ of an inch. 

t So many that no part of the building may be more th an 30 or 40 feet from one. 


Book V. 


OF ELECTRICITY. 


“Principles of Electricity,” by Charles Viscount Mahon, now Earl Stanhope. To the same w ork, 
the reader must be referred for an account of a discovery made by his lordship in the science of elec¬ 
tricity, which he denominated the “ returning stroke ,” by which, he asserts, that persons may be killed, 
and other vast mischief ensue by lightnings at the distance of several miles from the flash. It is proper 
also to observe that several respectable electricians, though willing to admit the fact as discovered 
by Earl Stanhope, yet do not seem to think that the danger attending the returning stroke can ever be 
great or formidable. See Cavallo’s Elect. Vol. ii. and iii. Morgan’s Lectures on Elect. Vol. ii. Dr. 
Hutton’s Diet. Art. Returning Stroke. 

PROP. XXV. The electric fluid passes easily through a vacuum. 

The air being a non-conductor, in proportion as it is removed, the effort of the electric fluid 
on the surface of the body positively electrified to pass to the next conductor, meets with less 
resistance, and therefore is diffused over a greater space. 

Exp. 1. Let a jar be charged in vacuo. 

2. Let a luminous conductor be placed in the circuit, and observe the fluid passing through it. 

3. Let a vacuum he made a part of the circuit in discharging a phial. 

4. Make a vacuum in a double barometer, and let the fluid pass from one leg to the other by con¬ 
necting one of the vessels of mercury with the conductor. 

5. The electric fluid may be made to pass through a large tube three feet in length, and four or 
five inches in diameter, if, being well exhausted, one end of it be connected with a large conductor.— 
The preceding experiments are to be performed in a dark room. 

Schol. 1. From the resemblance between these electrical appearances, and the atmospherical phe¬ 
nomena of the Aurora Borealis , meteors, &c. it is inferred, that these phenomena are produced by the 
electric fluid. 

Schol. 2. The success of the foregoing experiments depends, it is highly probable, upon the air in 
the jar, tube, &c. being rarefied in a high degree ; for Mr. W. Morgan, a gentleman deeply skilled in 
calculations and political arithmetic, has shewn that a perfect vacuum is absolutely impermeable to the 
electric fluid. See Phil. Trans, vol. lxxv. 

PROP. XXVI. Some fishes have the property of giving shocks analogous to those 
of artificial electricity ; namely, the Torpedo, the Gymnotus electricus, and the Silurus 
electricus. 

If the torpedo, whilst standing in water, or out of water, but not insulated, be touched with one 
hand, it generally communicates a trembling motion or slight shock to the hand. If the torpedo be 
touched with both hands at the same time, one hand being applied to its under, and the other to its 
upper surface, a shock will be received exactly like that occasioned by the Leyden Phial. When the 
hands touch the fish on the opposite surfaces, and just over the electric organs, then the shock is the 
strongest; but no shock is felt, if both hands are placed upon the electric organs of the same surface; 
which shows that the upper and lower surfaces of the electric organs are in opposite states of elec¬ 
tricity, answering to the plus and minus sides of a Leyden Phial. 

The shock given by the torpedo, when in air. is about four times as strong as when in water; and 
when the animal is touched on both surfaces by the same hand, the thumb being applied to one surface, 
and the middle finger to the opposite, the shock is felt much stronger than when the circuit is formed 
by both hands. 

This power of the torpedo is conducted by the same substances which conduct electricity, and is 
interrupted by those substances which are non-conductors of electricity. A circuit may be made of 
several persons joining hands, and the shock will be felt by them all at the same time; but the shock 
wi 11 not pass through the least interruption of continuity, not even the distance of the two hundreth part 
of an inch. 

No electric attraction or repulsion could be ever observed to be produced by the torpedo, nor 
indeed by any of the electric fishes. The shocks of the torpedo seem to depend' on the will of the 
animal. 

The gymnotus electricus, or electric eel, possesses all the electrical properties of the torpedo, 
but in a superior degree. When small fish are put into the water wherein the gymnotus is kept, they 
are generally stunned or killed by the shock, and then they are swallowed, if the animal be hungry. 

The strongest shock of the gymnotus will pass a very short interruption of continuity in the circuit. 
When the interruption is formed by the incision made by a penknife on a slip of tin-foil that is pasted 
on g-lass, and that slip is put into the circuit, the shock, in passing through that interruption, will show 
a small but vivid spark, plainly to be seen in a dark room. 


100 


OF ELECTRICITY. 


Book V. 


The gymnotus seems also to be possessed of a sort of new sense , by which he knows whether the 
bodies presented to him are conductors or not. This fact was ascertained by a great number of experi¬ 
ments, made by Mr. Walsh. 

The silurus electricus is known to have the power of giving the shock, but we have a very im¬ 
perfect account of its properties. 

A fourth electrical fish was found on the coast of Johanna, one of the Comora islands, in lat. 12° 13' 
south, by William Patterson ; and an account of it was published in the 76th vol. of the Phil. Trans. 

Schol. 1. When electricity is strongly communicated to insulated animal bodies, the pulse is quicken¬ 
ed, and perspiration increased; and if they receive, or impart electricity on a sudden, a painful sensa¬ 
tion is felt at the place of communication. But what is more extraordinary is, that the influence of the 
brain and nerves upon the muscles seems to be of an electric nature. 

We are indebted for this discovery to M. Galvani, a learned Italian, who has denominated that part 
of science, Animal Electricity. We shall, without pretending to enter at large on the subject, give the 
result of the principal observations hitherto made, together with three or four illustrative experiments. 

1. The nerve of the limb of an animal being laid bare, and surrounded with a piece of tin-foil, if 
a communication be formed between the nerve thus armed, and any of the neighbouring muscles, by 
means of a piece of zinc, strong contractions will be produced in the limb. 

2. If a portion of the nerve which has been laid bare be armed as above, contractions will be produc¬ 
ed as powerfully, by forming the communication between the armed and bare part of the nerve, as be¬ 
tween the armed part and muscle. 

3. A similar effect is produced by arming a nerve, and simply touching the armed part of it with 
the metallic conductor. 

4. Contractions will take place if a muscle be armed, and a communication be formed by means of 
the conductor between it and a neighbouring nerve. The same effect will be produced if the commu¬ 
nication be formed between the armed muscle and another muscle which is contiguous to it. 

5. Contractions may be produced in the limb of an animal by bringing the pieces of metal into con¬ 
tact with each other at some distance from the limb, provided the latter make part of a line of commu¬ 
nication between the two metallic conductors. 

6. Contractions can be produced in the amputated leg of a frog, by putting it into water, and 
bringing the two metals into contact with each other at a small distance from the limb. 

7. The influence which has passed through, and excited contractions in one limb, may be made to 
pass through and excite contractions in another limb. 

8. The heart is the only involuntary muscle, in which contractions can be excited by these experi¬ 
ments. 

9. Contractions are produced more strongly, the farther the coating is placed from the origin of 
the nerve. 

10. Animals which were almost dead have been found to be considerably revived by exciting this 
influence. 

11. When these experiments are repeated upon an animal that has been killed by opium, or by 
the electric shock, very slight contractions are produced; and no contractions whatever will take 
place in an animal that has been killed by corrosive sublimate, or that has been starved to death. 

12. Zinc appears to be the best exciter when applied to gold, silver, molybdena, steel, or copper. 
The latter metals, however, excite but feeble contractions when applied to each other. Next to zinc, 
in contact with these metals, tin and lead, and silver and lead, appear to be the most powerful exciters. 

Exp. 1. Place the limb of an animal, a frog for instance, upon a table; hold with one hand the 
principal nerve previously laid bare, and in the other hold a piece of zinc; let a small plate of lead 
or silver be then laid upon the table, at some distance from the limb, and a communication be formed, 
by means of water, between the limb and the part of the table where the metal is lying. If now, the 
silver be touched with zinc, contractions will be produced in the limb the moment that the metals 
come into contact with each other. The same effect will be produced, if the two pieces of metal be 
previously placed in contact, and the operator touch one of them with his finger. 

2. Let two amputated limbs of a frog be taken ; let one of them be laid upon a table, and its foot 
be folded in a piece of silver; let a person lift up the nerve of this limb with a silver probe, and an¬ 
other person hold in his hand a piece of zinc, with which he is to touch the silver including the foot; 
let the person holding the zinc in one hand, catch with the other the nerve of the second limb, and 
he who touches the nerve of the first limb is to hold in the other hand the foot of the second ; let the 
zinc now be applied to the silver, including the foot of the first, and contractions will be immediately 
excited in both limbs. 

3. Take a living flounder, lay it flat in a pewter plate, or upon a sheet of tin-foil, and put a piece 
«ff silver, as a shilling or half a crown, upon the fish. Then by means of a piece of metal, complete 


Book V. 


OF ELECTRICITY. 


101 


the communication between the pewter plate, or tin-foil, and the silver piece, on doing which the 
animal will give evident tokens of being affected. 

4. Let a person lay a piece of zinc upon his tongue, and a half crown, or other silver, under it ; 
on forming a communication between those two metals, by bringing their two edges into contact, he 
will perceive a peculiar sensation, a kind of cool, sub-acid taste, not exactly like, and yet not much differ¬ 
ent from that produced by artificial electricity. See Cavallo’s Elect. Vol. iii. 

Schol. 2. Electricity has been administered for various diseases. Mr. Cavallo has taken great pains 
in ascertaining the cases in which electricity has been successfully applied. We are informed by that 
gentleman, that rheumatic disorders , even of long standing, are relieved, and generally quite cured. 
Deafness, the toothach, swellings in general, inflammations of every sort, palsies, ulcers, cutaneous 
eruptions, the St. Vitus’ dance, scrofulous tumours, cancers, abscesses, nervous headach, the dropsy, 
gout, agues, and obstructions, have all been considerably relieved, and in many instances perfectly 
cured, by the application of electricity. A full account of the method of administering electricity in 
the cases above mentioned, with an accurate description of the instruments used, may be seen in the 
2d Vol. of Cavallo’s Complete Treatise of Electricity. 

PROP. XXVII. There is a considerable analogy and difference between mag¬ 
netism and electricity. 

The power of electricity is of two sorts, positive and negative; bodies possessed of the same sort 
of electricity, repel each other, and those possessed of different sorts attract each other. In magnet¬ 
ism, every magnet has two poles; poles of the same name repel each other, and the contrary poles 
attract each other. 

In electricity, when a body in its natural state is brought near to one electrified, it acquires a con¬ 
trary electricity, and becomes attracted by it. In magnetism, when a ferruginous substance is brought 
near to one pole of a magnet, it acquires a contrary polarity, and becomes attracted by it. 

One sort of electricity cannot be produced by itself. In like manner, no body can have one mag¬ 
netic pole without the other. 

The electric virtue may be retained by electrics, but it easily pervades non-electrics. The mag¬ 
netic virtue is retained by ferruginous bodies, but it easily pervades other bodies. 

On the contrary, the magnetic power differs from the electric, in that it does not affect the senses 
with light, smell, taste, or noise, as the electric does. 

Magnets attract only iron, whereas the electric power attracts bodies of every sort. 

The electric virtue resides on the surface of electrified bodies, but the magnetic is internal. 

A magnet loses nothing of its power by magnetising other bodies, but an electrified body loses part 
'jf its electricity by electrifying other bodies. See Cavallo’s Magnetism, Part n. Chap. 2. 


BOOK VI, 


OF OPTICS; OR, THE LAWS OF LIGHT AND VISION. 
CHAPTER. I. 


Plate 6, 
Fig. 1. 

Fig. 2. 


Of Light. 

Def. I. IilGHT is that which, proceeding from any body to the eye, produces 
the perception of seeing. 

Def. II. A Ray of Light is any exceedingly small portion of light as it comes from 
a luminous body. 

Def. III. A body, which is transparent, or affords a passage for the rays of light, 
is called a Medium. 

Def. IV. Rays of light which, coming from a point, continually separate as they 
proceed, are called Diverging Rays. 

Def. V. Rays which tend to a common point, are called Converging Rays. The 
divergency, or convergency, of rays, is measured by the angle contained between the 
lines which the rays describe. 

Def. VI. Rays of light are parallel, when the lines which they describe are parallel. 

Def. VII. A Beam of light is a body of parallel rays ; a Pencil of rays, is a body 
of diverging or converging rays. 

Def. VIII. The point, from which diverging rays proceed, is called the radiant 
point ; that, to which converging rays are directed, is called the focus. 

If the rays proceed from B ; BD, BA, BC, BE, are diverging rays, and B is the radiant; if the rays 
tend toward B ; DB, AB, &c. are converging rays, and B is the focus. J 

If the rays AC, BC, converge to the focus C, passing on from thence in a right line, they become 
diverging, and C becomes a radiant. 

Def. IX. A ray of light, bent from a straight course in the same medium, is said 
to be inflected. 

PROPOSITION I. 


Rays of light consist of particles of matter. 

For, like all matter with which we are acquainted, they are capable of being inflected out of their 
course by attraction. 

Exp. 1. If a beam of light be admitted into a dark room through a small hole, and the edge of a knife 
be brought near the beam, the rays, which would otherwise have been in a straight line, will be inflect¬ 
ed toward the knife. The edge of any other thin plate of metal &c. produces the same effect. 

2. The shadow of a small body, as a hair, a thread, &c. placed in a beam of the sun’s light, will be 
much broader than it ought to be if the rays of light passed by these bodies in right lines. 

3. A beam of light passing through an exceedingly narrow slit, not above p ar t 0 f aQ j nc b broad, 
will be split into two, and leave a dark space in the middle. 



Chap. I. 


OF LIGHT. 


103 


PROP. II. Every visible body emits particles of light from its surface in all direc¬ 
tions, which, passing without obstruction, move in right lines. 

Wherever a spectator is placed with respect to a luminous body, every point of that part of the sur¬ 
face which is turned toward him is visible to him; the particles of light are, therefore, emitted in all 
directions, and those rays only are intercepted in their passage by an interposed object, which would be 
intercepted upon the supposition that the rays move in right lines. 

Exr. 1. Let a portion of a beam of light be intercepted by any body, the shadow of that body will be 
bounded by right lines passing from the luminous body, and meeting the lines which terminate the 
opaque body. 

2. A ray of light, passing through a small orifice into a dark room, proceeds in a straight line. 

3. Rays will not pass through a bended tube. 

Schoo. Rays of light are properly represented by right lines. 

PROP. III. The rays of light move with great velocity. 

The velocity of light is much greater than that of sound ; for the flash of a gun, fired at a considera¬ 
ble distance, is seen some time before the report is heard. The clap of thunder is not heard till some 
time after the lightning has been seen. 

This proposition is proved by observations made on the satellites of the planet Jupiter, and on the 
aberration of the rays of light from the fixed stars, as will be shown in treating upon Astronomy; from 
whence it will be seen, that the velocity is at the rate of 200,000 miles in one second of time. 

PROP. IV. The particles of light are exceedingly small. 

Otherwise their velocity would render their momentum too great to be endured by the eye without 
pain. 

Exp. 1. If a candle be lighted, and there be no obstacle to obstruct the progress of its rays, it will fill 
all the space within two miles every way, before it has lost the least sensible part of its substance. 

2. Rays of light will pass without confusion through a small puncture in a piece of paper, from sev¬ 
eral candles in a line parallel to the paper, and form distinct images on a sheet of pasteboard placed be¬ 
hind the paper. 

PROP. V. The quantities of light, received from a luminous body upon a given 
surface, are inversely as the squares of the distances of the surface from the luminous 
body. 

Let ABD, EFG, be two concentric spherical surfaces; of which let ELFI, AHBK, be two similar Plate 6. 
portions. Let the rays CE and CF, with the rest proceeding from the centre C, fall upon the portion Fig. 3. 
ELFI, and cover it; it is evident from inspection, that the same rays at the distance CH will cover the 
portion AHBK only; now these rays being the same in number at each place, will be as much thinner 
in the former, than they are in the latter, as ELFI is larger than AHBK; but these spaces being simi¬ 
lar portions of the surfaces of spheres, have the same ratio to each other, that the surfaces themselves 
have ; that is, they are to each other as the squares of their radii CL, CH; the density of the rays is 
therefore inversely as the squares of these radii, or of their distances from the luminous point C. 

Exr. The light, passing from a candle through a square orifice, will diverge as it proceeds, and will Plate 12. 
illuminate surfaces which will be to each other as the squares of their distances from the candle. Thus Fl S' 
at the distance AF the candle will illuminate the square BF; at the distance AO it will illuminate the 
surface CO equal to four times BF, and at the distance AS it will illuminate the surface DS equal to 
nine times BF ; but AF, AO, and AS, are as 1, 2, and 3; consequently the illuminated surfaces are as the 
squares of the distances. 

PROP. VI. If the distance between rays diverging from different radiant points be 
the same, the distances of the radiant points are inversely as the divergency of the 
rays. 

Let D and E be two different radiants; and let the rays diverging from D describe the lines DA, piate 6. 
DB, and the rays diverging from E describe the lines EA, EB; so that, at the points A and B, the dis- Fig. 4. 
tance between the former rays shall be the same with the distance between the latter, and let EC, DC, 
be the perpendicular distances of the radiants E, D. At the point E make the angle ZEC equal to 
ADC, which is half ADB ; whence ZEC and ADC (El. V. 7.) have the same ratio to AEC. But if 
these angles are small, they are very nearly in the proportion of their tangents ZC, AC, And because 
the angle ADC is equal to the angle ZEC (El. I. 28.) AD is parallel to ZE; and because these lines are 


104 


OF OPTICS. 


Book VI. 


Plate 6. 
Fig. 4. 


Plate 6. 
Fig. 6. 


Plate 6. 
Fig. 5. 


parallel, (El. I. 290 the angles CAD, CZE, are equal; whence the two triangles ZEC, ADC, are equi¬ 
angular, and (El. VI. 4.) EC is to DC, as ZC to AC, or (from what was shown above) as ADC to AEC ; 
that is, the distance of the radiant E is to the distance of the radiant D, as half the angle of divergency 
of the rays which proceed from D is to half the divergency of the rays which proceed from E, or as the 
whole angle of divergency ADB to the whole angle of divergency AEB ; that is, the distances of the 
radiants are inversely as the divergency of the rays. 

PROP. VII. If the distance between converging rays tending to different foci be 
the same, the distances of the foci are inversely as the convergency of the rays. 

Let AD, BD, be lines described by rays converging to the focus D, and AE, BE, lines described by 
other rays converging to E, and let the distance AB, at the points A and B, be the same between 'the 
former and the latter rays. The angles ADB, AEB, are in this case the angles of convergency; and 
EC, DC, are distances of the foci to which they respectively tend. Now it was proved in the last 
Prop, that EC is to DC as ADB is. to AEB. Therefore the distances of the foci are inversely as the 
convergency of the rays. 

PROP. VIII. If rays proceed from a radiant at an infinite distance, their diver¬ 
gency is considered as nothing, and the rays are considered as parallel. 

Since (by Prop. VI.) the divergency of rays is inversely as the distance of the radiant, when the dis¬ 
tance of the radiant is infinitely great the angle of divergency is infinitely small, and the rays may be 
considered as parallel. 

Cor. Hence all the rays which come from the centre, or any other given point, of the sun’s surface, 
are considered as parallel. 

PROP. IX. If rays tend to a focus at an infinite distance, their convergency is con¬ 
sidered as nothing, and the rays are considered as parallel. 

Since (by Prop. VII.) the convergency is inversely as the distance of the focus, when that distance 
is infinitely great, the angle of convergency is infinitely small. 


CHAPTER II. 

Of Refraction . 

SECT. I. 

OF THE LAWS OF REFRACTION. 

Def. X. A ray of light bent from a straight course by passing out of one medium 
into another, is said to be refracted. 

Def. XI. The Angle of Incidence is that, which is contained between the line de¬ 
scribed by the incident ray, and a line perpendicular to the surface on which the ray 
strikes, raised from the point of incidence. 

Def. XII. The Angle of Refraction is that, which is contained between the line de¬ 
scribed by the refracted ray, and a line perpendicular to the refracting surface at the 
point in which the ray passes through that surface. 

Def. XIII. The Angle of Deviation is that which is contained between the line of 
direction of an incident ray, and the direction of the same ray after it is refracted. 

AC is a ray of light; HK the surface of the refracting medium; CF the refracted ray; OP the per¬ 
pendicular; ACO the angle of incidence ; PCF the angle of refraction; and FCL the angle of devi¬ 
ation. 

Schol. The radiant point and focus may be either real or imaginary. If the rays r re, r o, diverging 
from the radiant r, suffer refraction and move on in the directions of the lines n A, o B, which produced 
in the contrary direction would meet in R, this radiant point is imaginary. 

If the rays Ip, Lq , tending toward the point F, be refracted atp and q , and acquire a direction to¬ 
ward /, the focus F is imaginary. 



C$AP. II. 


OF REFRACTION. 


105 


PROP. X. The attracting force of any medium, acting upon a ray of light, is every 
where perpendicular to the refracting surface. 

If the medium be uniform in all its parts, its immediate power upon the ray of light will be equally Plate a. 
strong in every point of a plane drawn parallel to the refracting surface ; though its strength may be Fig. S. 
different in the next parallel plane, and so onward as far as that power is extended on each side of the 
surface of the medium. The extent of this power will therefore be terminated by two planes, parallel 
to each other and to the refracting surface. Let lx be a particle of light, acted upon by the refractive 
power of the medium whose refracting surface is DC. It is evident that the refractive power at O will 
move the particle R in the direction RO ; and taking any two points D, C, at equal distances on each 
side of O, the powers at D and C being equal, and acting at equal distances, RD, RC, equally inclined 
to RO, cannot move R in any direction but that of RO. The same may be shown of the powers at every 
point of the line DC, and in every line parallel to DC, that is, of the whole power of the medium. 

PROP. XI. A ray of light, in passing out of a rarer into a denser medium, is re¬ 
fracted toward a perpendicular to the surface of the denser, raised from the point in 
which the ray meets the medium ; in passing out of a denser into a rarer medium, it 
is refracted from the same perpendicular. 

Let a ray of light, AC, pass obliquely out of a rarer medium X, into a denser medium Z ; let HK be Plate 6. 
the plane surface of the denser medium; from the point C, in which the ray AC passes into the denser Fig. 6. 
medium, raise the perpendicular OCP; the ray will be refracted out of the direction ACL, toward the 
perpendicular OCP. 

Because the ray is more attracted by the denser medium than by the rarer, it will be accelerated 
on entering the medium Z; for whilst the ray is so near the surface of the medium Z as to be within its 
attraction, and more attracted toward the denser than toward the rarer, this attraction conspires with 
the motion of the ray, and, consequently, increases its velocity. And, since the action of the attracting 
force of the medium Z, must (by Prop. X.) be in the direction of a line OCP perpendicular to its sur¬ 
face, if the oblique motion of the ray in the direction AC be resolved into two others, AD parallel to 
the surface HK, and AB, or DC, perpendicular to it, the parallel motion AD cannot be accelerated or 
retarded by the attraction which acts in the direction OC; the change of velocity, therefore, which the 
ray receives from the attracting force, must he made in the perpendicular part of its motion DC. Take 
CG greater than DC representing the perpendicular motion of the ray after passing into the denser me¬ 
dium ; and take CE equal to AD representing the parallel part of the motion of the ray, which, because 
it is parallel to AB, remains the same when the ray enters the denser medium. The ray, therefore, at 
its entering the medium Z, may be considered as acted upon by two forces CE, CG, and consequently 
(Book II. Prop. XIV.) will describe CF the diagonal of a parallelogram, the sides of which are CE, CG. 

Now, of these sides, CE remaining the same, whilst CG becomes greater than CD, the angle GCF (from 
the nature of the parallelogram) will be less than the angle NCL, equal (El. I. 15.) to ACD. Therefore 
the ray, after it has passed into the denser medium, makes a less angle with the perpendicular OCP 
than AC, the ray before it passes into the denser medium ; that is, the ray, in passing out of the rarer 
into the denser medium is refracted toward the perpendicular. On the contrary, whilst the ray of light 
FC is passing out of the denser medium Z into the rarer medium X, it is more attracted by the denser 
than by the rarer medium, and is therefore more drawn toward the former than toward the latter ; 
whence the attraction opposes the motion of the ray, and will retard it as much, as iu passing out of the 
rarer into the denser medium it was accelerated ; and, consequently, the effect will be the reverse of 
that which was shown in the former case. 

Schol. 1. Although there is no doubt that refraction is performed gradually, and in time, during 
which the light really describes a curve line extending quite through the refracting space, and connect¬ 
ing the refracted with the incident ray, which are tangents to this curve at its respective extremities, 
yet both the time and space are so small, that experiment has never been able to render even the space 
perceptible, so that the incident and refracted rays are commonly considered as forming a perfect angle 
precisely at the surface separating the two mediums. 

Schol. 2. The principles of optics are demonstrated upon the supposition that light-is a homogeneal 
substance ; and though light will appear to be compounded of several kinds of rays, yet the principles 
of refraction, reflection, &c. are mathematically true when applied to rays of any one sort. 

Exp. 1. Let a perpendicular cylindrical vessel he so placed that the sun, shining upon its side NA, Plate 
may cast a shadow of the side to a point L in the bottom of the vessel. This shadow is terminated by Fig. 7. 
SNL, a ray which passes, in a right line, by the edge of the vessel. If the vessel he filled with water, 
the shadow will recede, as the water is poured into the vessel, from the point L, which terminated it 

14 


106 


OF OPTICS. 


Book VL 


when the vessel was empty, toward the side NA, on which the sunshines, and will be terminated by the 
the ray ONC ; that is, the ray SNL, which first terminated the shadow, by passing out of the air into 
the water, is refracted toward AN, a line drawn perpendicular to the surface of the water at the point 
in which the ray enters the water; or the angle of refraction is less than the angle of incidence. 

2. Let a small bright object be laid upon the bottom of a cylindrical vessel NBAL at C. Let the 
spectator’s eye be so placed at S, as just to lose sight of the object at C; that is, so that a ray passing in 
a right line from the remote edge of the object toward the eye at S will be intercepted by the edge of 
the vessel, or that the first ray which is not intercepted will pass in the direction ONC above the eye. 
Whilst the eye continues in the same situation, if the vessel be filled with water, the object will become 
visible ; that is, the ray which passed from fhe remote edge of the object, in a right line CNO, by the 
vessel, in entering the air is refracted into the direction NS, toward the eye, or from the perpendicular 
PNA. 

PROP. XII. All refraction is reciprocal. 

Plate 6. The ray AC, in passing out of the medium X into Z, is refracted into CF,‘because it is accelerated 

F'g- 6- at its entrance into Z by the the greater attraction of the denser medium ; and the ray FC in passing out 
of Z into X is refracted into CA, because it is retarded by the same attraction. Since, then, the accelera¬ 
tion and retardation are produced by the same degree of attraction in opposite directions they will be 
equal to one another, and the refractions produced by them will be equal, but in opposite directions; that 
is, if the refracted ray becomes the incident ray, the incident ray will become the refracted one ; or, 
the refractions are reciprocal. 

PROP. XIII. In any two given mediums, the sine of any one angle of incidence has 
the same ratio to the sine of the corresponding angle of refraction, as the sine of any 
other angle of incidence has to the sine of its corresponding angle of refraction. 

Plate 16,. Let AC represent the velocity of a ray of light obliquely incident on the plane surface FG of a denser 

Fig. 1. medium W at the point C. Resolve this motion, which is constant and invariable in the same medium, 
into AB perpendicular to the refracting surface and BC parallel to it. Of these (as was shown in Prop. 
XI.) only the perpendicular motion AB is accelerated, the parallel motion BC continuing unaffected bv 
the attraction. Next (since most probably the intensity of the refractive power is greatest precisely at 
the refracting surface, and gradually diminishes on each side to the limits of its action, thus producing a 
variable acceleration,) suppose the refracting space to be divided into strata by planes parallel to the 
refracting surface at the point through which the ray passes, and so near to each other that the refrac¬ 
tive power, and, of course, the acceleration may be considered uniform between them. Now (B. II. 
Prop. XXVI. Cor. 4.) the space passed through by a motion uniformly accelerated is proportional to the 
difference of the squares of the initial and final velocities. But the space passed through by the perpen¬ 
dicular part of the motion of any ray while passing through the same stratum, is always the same, 
(that is, the thickness of the stratum) whatever the obliquity of the ray may be; therefore the dif¬ 
ference between the square of the perpendicular velocity of any ray at its entering any one stratum 
and the square of its perpendicular velocity at its leaving the same 'stratum is the same as the dif¬ 
ference between the squares of the perpendicular velocities of any other ray at its entering and leaving 
the same stratum, however different the obliquity, and, of course, the actual perpendicular velocities 
of the rays may be. And as the number of these differences, with reference to any single ray, is 
equal to the number of strata assumed, the sums of an equal number of equal differences must be equal. 
Therefore the whole differences produced in passing through all the strata or whole refracting space, 
or the difference between the squares of the perpendicular velocities in one medium until the rays en¬ 
ter the refracting space, and in the other after emerging from it, is always the same, however the actual 
perpendicular velocities may vary with the obliquity of incidence, and however the refracting force 
may vary during refraction. Take CD = BC to represent the parallel motion of the ray after refrac¬ 
tion, as that continues unaltered, and draw DE in the denser medium perpendicular to the surface to 
represent the perpendicular velocity of the refracted ray. The square of DE must exceed the square 
of AB by a certain quantity which continues constant in all positions of the ray AC. Now since 
€D 3 =BC 2 , and DE 2 exceeds AB 1 by a constant quantity, CD 2 -fDE 2 or CE 4 must exceed BC 4 -f- 
AB 8 or AC 2 by the same constant quantity. Therefore as AC and AC 2 are constant quantities, so CE 4 and 
CE are also constant; and as AC expresses the direct velocity of fhe incident ray which never varies 
with the angle of incidence, so CE expresses the direct velocity of the refracted ray, which is the same at 
all angles. On the centre C, and with CB or CD, representing the parallel velocity, as a radius, describe 
a circle ; it is plain that AC is the secant of the angle ACB or cosecant of the angle of incidence, and 
CE the secant of the angle DCE or cosecant of the angle of refraction ; and as these are constant 
quantities, however the radius or parallel motion may vary, the coescants of incidence and refraction 


Chap. II. 


OF REFRACTION. 


107 


in the same two mediums have a constant ratio to each other; but sines are inversely as the cosecants 
of the same angles, therefore the sines also have always the same ratio to each other. When light 
passes from a denser medium into a rarer, the same reasoning may easily be adapted to the retardation 
—the differences of the squares both of the perpendicular and direct velocities of the rays before and 
after refraction being the same at all angles of inclination, and the direct velocity of the refracted ray 
being less than that of the incident ray by a certain constant quantity, &c. the terms of the ratio being- 
reversed. 

If a, 6, d , e, be read instead of A, B, D, E, respectively, the figure exhibits an example of a much 
greater angle of incidence. 

Cor. 1. Hence, when the angle of incidence is increased, the corresponding angle of refraction will 
also be increased ; because the ratio of their sines cannot continue the same, unless they be both in¬ 
creased ; and if two angles of incidence be equal, the angles of refraction will also be equal. 

Cor. 2. Hence the angle of deviation varies with the angle of incidence. 

Schol. 1. If a ray of light, AC, pass obliquely out of air into water, AD, the sine of the angle of in¬ 
cidence ACD is to NS, the sine of the angle of refraction NCF, nearly as 4 to 3; therefore, supposing 
the sines proportional to the angles, the sine of FCL, the angle of deviation, is as the difference between 
AD and NS, that is, as 4 — 3, or 1; whence the sine of incidence is to the sine of the angle of devia¬ 
tion as 4 to 1. In like manner it may be shown, that, when the ray passes obliquely out of water into 
air, the sine of the angle of incidence will be to that of deviation, as NS to AD — NS, that is, as 3 to 1. 
In passing out of air into glass, the sine of the angle of incidence is to that of refraction, as 3 to 2, and 
to that of deviation, as 3 to 3 — 2, or 1 ; and in passing out of glass into air, the sine of the angle of 
incidence is to that of refraction, as 2 to 3, and to that of deviation as 2 to 1. 

Cor. 3. Hence a ray of light cannot pass out of water into air at a greater angle of incidence than 
48° 36', the sine of which is to radius as 3 to 4, Out of glass into air the angle must not exceed 40° 11', 
because the sine of 40° 11' is to radius as 2 to 3 nearly; consequently, when the sine has a greater pro¬ 
portion to the radius than that mentioned, the ray will not be refracted. 

Schol 2. It must be observed, that when the angle is within the limit, for light to be refracted, some 
of the rays will be reflected. For the surfaces of all bodies are for the most part uneven, which occa¬ 
sions the dissipation of much light by the most transparent bodies ; some being reflected, and some re¬ 
fracted, by the inequalities on the surfaces. Hence a person can see through water, and his image re¬ 
flected by it at the same time. Hence also, in the dusk, the furniture in a room may be seen by the re¬ 
flection of a window, while objects that are without are seen through it. 

Exp. Upon a smooth board draw a right line BCD, and on its extremities erect the perpendiculars 
BA and DE in opposite directions ; on the middle C, of line BD, as a centre, and with any extent greater 
than BC, intersect the line BA suppose in A, and with an extent greater than CA in the ratio of 4 to 3, 
intersect the line DE, perhaps in E. Then if pins be stuck perpendicularly at A, C, and E, and the 
board be dipped in the water as far as the line BD, the pin at E will appear in the same line with the 
pins at A and C. This shows, that the ray which comes from the pin E is so refracted atC, as to come 
to the eye along the line CA ; whence the sine of incidence is to the sine of refraction as 4 to 3. If 
other pins were fixed along CE, they would all appear in AC produced; -> Ich shows that the ray is 
bent at the surface only. The same may be shown, at different inclination f the incident ray, by means 
of two moveable rods turning upon the centre C, which always keep the ratio of the sines, as 4 to 3. 
Also the sun’s rays, coinciding with AC, may be shown to be refracted in the same manner. 

PROP. XIV. Rays of light, which pass perpendicularly out of one medium into 
another, suffer no refraction. 

When AC, the incident ray, concides with OC, the perpendicular, the action of the medium Z or X 
to accelerate or retard the motion of the ray, being perpendicular to its surface, cannot turn the ray 
out of its perpendicular path. 

PROP. XV. When parallel rays pass obliquely out of one medium into another 
through a plane surface, they will continue parallel after refraction. 

Let AB, CD, be parallel rays, falling on the plane surface RBD of a medium of different density ; 
because they make equal angles of incidence with their respective perpendiculars OP, ST, they will 
suffer an equal degree of refraction; that is, the angles of refraction EBP, FDT, will be equal; whence 
the refracted rays BE, DF, will be parallel. 

PROP. XVI. Through a plane surface, if diverging rays pass out of a rarer into 
a denser medium, they are made to diverge less ; and if they pass out of a denser into 


Plate 6. 
Fig. 6. 


Plate 16 
Fig. 1. 


Plate 6, 
Fig. 6. 


Plate 6. 
Fig. 9 


108 


OF OPTICS. 


Book. VI. 


Plate 4. 
Fjg. 10. 


Fig. 11. 


Plate 6. 
Sig 13. 


a rarer medium, they are made to diverge more: If converging rays pass out of a 
rarer into a denser medium, they will be made to converge less; it out of a denser in¬ 
to a rarer, to converge more. 

Let the diverging rays AB, AE, AF, pass out of a rarer into a denser medium, through the plane 
surface GH, and let the ray AB be perpendicular to that surface ; the rest being refracted toward their 
respective perpendiculars IK, LM, and that the most which falls the farthest from B, they will proceed 
in the directions EN and FO, divergingin a less degree from the ray AP, than they did before refraction ; 
whereas, had they proceeded out of a denser into a rarer medium they would have been refracted from 
their perpendiculars EK, FM, and those the most which were the most oblique, and therefore would 
have diverged more than before. Again, let the converging rays AB, CD, EF, pass out of a rarer into 
a denser medium, through the plane surface GH, and let the ray AB be perpedicular to that surface ; 
the other rays being refracted toward their respective perpendiculars IK, LM, and EF being refracted 
more than CD, they will proceed in the directions DN, FN, converging in a less degree toward the ray 
AN, than they did before ; whereas, had the first medium been the denser, they would have been re¬ 
fracted the other way, and therefore have converged more. 

Def. XIV. A Lens is a round piece of polished glass, which has both its sides 
spherical, or one spherical and the other plane. 

A lens may either be convex on both sides, plano-convex, concave, plano-concave, or convex on one 
side and concave on the other; which last is called a meniscus. 

In plate 6. rig. 12. sections of these, formed by a plane passing perpendicularly through their centres, 
are represented. 

Def. XV. The Axis of a Lens is a right line passing through its centre, perpen¬ 
dicular to both its surfaces, and the extremities of the axis are its poles. 

Each kind of lens is generated by the revolution of a section of the lens about this line. Thus, in the 
first lens, if c c 6, adb, revolve about c c?, the convex lens will be formed. 

Def. XVI. In every beam of light, the middle ray is called the Axis. 

Def. XVII. Rays are said to fall directly upon a lens, if their axis coincides with 
the axis of the lens ; otherwise, they are said to fall obliquely. 

Def XVIII. The point, in which parallel rays are collected by passing through a 
lens, is called the Focus of parallel rays of that lens. 

PROP. XVII. Through a convex surface of the denser medium, parallel rays, pass¬ 
ing out of a rarer into a denser medium, will become converging ;—diverging rays will 
be made to diverge less, to become parallel, or to converge, according to the degree 
of divergency before refraction, or of the convexity of the surface;—rays converging 
toward the centre of convexity will suffer no refraction;—rays converging to a point 
beyond the centre of convexity will be made more converging ;—and rays converging 
toward a point nearer the surface than the centre of convexity, will be made less con¬ 
verging by refraction ;—and when the rays proceed out a denser into a rarer medium, 
through a concave surface of the denser, the contrary occurs in each case. 

Let AB, ID, be parallel rays entering a denser medium through the convex surface CDE, whose 
centre of convexity is L; and let one of these, ID, be perpendicular to the surface. This will pass on 
through the centre without suffering any reaction ; but the other, being oblique to the surface, will 
be refracted toward the perpendicular LB, and will therefore be made to proceed in some line, as PK. 
converging toward the other ray, and meeting it in K, the focus. Had one ray diverged from the otheri 
suppose in the line MP, it would, by being refracted toward its perpendicular LB, have been made 
either to diverge less, be parallel, or to converge. Let the line ID be produced to K ; and if the ray 
had converged, so as to have described the line NBL, it would then have been coincident with its per¬ 
pendicular, and have suffered no refraction. If it had proceeded with less convergency toward any point 
beyond L in the line IK, it would have been made to converge more by being refracted toward the 
■perpendicular LB, which converges more than it; and had it proceeded with more convergency than Bk. 


Chap. II. OF REFRACTION. 109 

that is, toward any point between Dand L, being refracted toward the perpendicular, it would have been 
made to converge less. 

And the contrary happens, when rays proceed out of a denser into a rarer medium, through a concave 
surface of the denser. For being now refracted from their respective perpendiculars, as they were be¬ 
fore toward them, if they are parallel before refraction, they diverge afterward; if they diverge, their 
divergency is increased ; if they converge in the direction of their perpendiculars, they suffer no refrac¬ 
tion ; if they converge less than their respective perpendiculars, they are made to converge still less, 
to be parallel, or to diverge; if they converge more, their convergency is increased. All which may 
clearly be seen by the figure, imagining the rays AB, ID, &c. bent the contrary way in their refractions 
to what they were in the former cases. 

Exp. Let parallel, diverging, and converging rays pass through a convex lens; the several Cases of 
this proposition will be confirmed. 

If CDEH be a convex lens, whose axis is IK, let L be the centre of the first convexity CDE, and 
M that of the other CHE; and let the ray AB be parallel to the axis; through B draw the line LN, 
which will be perpendicular to the surface CDE at that point. The ray AB in entering the denser 
substance of the lens will be refracted toward the perpendicular, and therefore proceed after it has en¬ 
tered the surface at B in some direction inclined toward the axis, as BP. Through M the centre of 
convexity of this surface and the point P draw the line MR, which passing through the centre will 
be perpendicular to the surface at P, and the ray now entering a rarer medium will be refracted 
from the perpendicular into some direction as PK. In like manner, and for the same reasons, the 
parallel ray ST on the other side the axis, and also all the intermediate ones, as XZ, &c. will meet it 
in the same point, unless the rays AB and ST enter the surface of the lens at too great a distance from 
the axis IK, the reason of which will be afterward explained. 

The point K where the parallel rays AB, ST, &c. are supposed to be collected by passing through 
the lens CE, is called the focus of parallel rays of that lens. 

If the rays come diverging from a point equally distant from the surface as the focus of parallel 
rays, they will be rendered parallel; if from a point farther from the surface than L, they will be 
brought to a point beyond L; if from a point nearer than L, they will diverge less ; as may be infer¬ 
red from Prop. XII. 

If the rays come converging toward L, they will suffer no refraction; if toward a point beyond L, 
they will become more converging; if toward a point nearer the surface than L, they will become less 
converging; as is sufficiently explained in the proof of this proposition. 

Schol. If the rays AB, CD, EF, be parallel to each other, but oblique to GH, the axis of the lens Plate 6. 
IK, or if the diverging rays CB, CF, proceed as from some point C, wffiich is not situated in the axis Fi g- 
of the lens, they will be collected into some point as L, not directly opposite to the radiant C, but 
nearly so ; for the ray CD, which passes through the middle of the lens, and falls upon the surface of 
it with some obliquity, will itself suffer a refraction at D and XT; but it will be refracted the contrary 
way in one place from that in the other; and these refractions will be equal in degree, if the sur¬ 
faces are parallel, as we may easily perceive if we imagine ND to be a ray passing out of the lens 
both at N and D, for it is evident the line ND has an equal inclination to each surface at both its ex¬ 
tremities. Upon which account the difference between the situation of the point L, and one.directly 
opposite to C, is so small, that it is generally neglected; and the focus is supposed to be in that line, 
in which a ray that would pass through the middle point of the lens, were it to suffer no refraction, 
would proceed. 

PROP. XVIII. When rays pass out of a rarer into a denser medium, through a con¬ 
cave surface of the denser, if the rays are parallel before refraction, they are made 
to diverge ;—if they are divergent, they are made to diverge more, to suffer no refrac¬ 
tion, or to diverge less, according as they proceed from some point beyond the centre, 
from the centre, or from some point between the centre and the surface;—if they are 
convergent, they are either made less converging, parallel, or diverging, according 
to their degree of convergency before refraction :—and the reverse, in, passing out 
of a denser into a rarer medium through a convex surface of the denser. 

Let MF, 01, be two parallel rays entering a concave and denser medium, the centre of ivhose con- Plate 
vexity is H, and the perpendicular to the refracting surface at the point F is LH; the ray 01, if we sup- Fig. 1 
pose it perpendicular to the surface, w ill proceed on directly without refraction, but the oblique ray MF, 
being refracted toward the perpendicular HL, will recede from the other ray 01. If the ray MF had 
proceeded from a point in 01 farther from the surface than H, it would have been bent nearer to the 


<3> Ci 


no or OPTICS. Book VI- 

perpendicular, and therefore have diverged more; if it had diverged from the centre H, it would have 
fallen in with the perpendicular HL, and not have been refracted at all; and had it proceeded from a 
point nearer the surface than the centre H, it would, by being refracted toward the perpendicular HL, 
have proceeded in some line nearer it than it otherwise would have done, and so would diverge less 
than before refraction. Lastly, if it had converged, it would have been rendered less converging, parallel, 
or diverging, according to the degree of convergency, which it had before it entered into the refracting 
surface. 

If the same rays proceed out of a denser into a rarer medium through a convex surface of the dens¬ 
er, the contrary happens in each supposition ; the parallel rays are made to converge ; those which 
diverge less than their respective perpendiculars, that is, those which proceed from a point beyond 
the centre, are made less diverging, parallel, or converging, according to the degree in which they 
diverge before refraction ; those which diverge more than their respective perpendiculars, that is, 
those which proceed from a point between the centre and the refracting surface, are made to di¬ 
verge still more. And those which converge, are made to converge more. All which may easily be 
seen by considering the situation of the rays with respect to the perpendicular HL. 

Exp. Let parallel, diverging, and converging rays pass through a concare lens; the several cases 
Plate 6. of this proposition will be confirmed; thus, let ABCD represent a concave lens, EO its axis, FH the 
Fig. 15. radius of the first concavity, IK that of the second; produce HF to L, and let MF be a ray of light 
entering the lens at the point F. This ray being refracted toward the perpendicular FL, will pass 
on to some point, as K, in the othersurface, more distant from the axis than F, and being there refract¬ 
ed from the perpendicular IK, will be diverted farther still from the axis, and proceed in the direction 
KN, as from some point O, on the first side of the lens. In like manner other rays, as PQ., parallel 
to the former, will proceed after refraction at both surfaces as from the same point O ; which upon 
that account will be the imaginary radiant of parallel rays of this lens. 

If the rays diverge before they enter the lens, their imaginary radiant is then nearer the lens than 
that of the parallel rays. If they converge before they enter the lens proceeding toward some distant 
point in the axis, as E, they are then rendered less converging: if they converge to a point at the 
same distance from the lens with the focus of parallel rays, they then go out parallel ; if to a point 
at a less distance, they remain converging, but in a less degree than before they entered the lens. 

Schol. If the lens is plane on one side, and convex or concave on the other, the refraction is sim¬ 
ilar, but in a less degree. In a meniscus, if he convexity on one side be equal to the concavity on the 
other, the two sides will produce equal and contrary effects, and the inclination of the rays to each 
other will be the same after refraction as before. If the convexity be greater than the concavity, the 
meniscus will have the effect of a lens which has its convexity equal to the excess of the convexity 
of the meniscus above its concavity ; and the reverse, if its concavity exceed its convexity. 

PROP. XIX. When diverging rays are made to converge by passing through a 
convex lens, as the radiant approaches toward the lens on one side, the focus departs 
from it on the other ; and the reverse. 

For, the nearer the radiant point is to the lens, the more the rays which fall upon the lens diverge 
before refraction ; whence (the power of the refracting medium being given) they will converge the 
less after refraction, and have their focal point at the greater distance from the surface : on the contra¬ 
ry, the more remote the radiant point is from the lens, the less the incident rays will diverge, and 
consequently the more will the refracted rays converge, and the nearer will the focus be to the sur¬ 
face, till, at an infinite distance of the radiant, the rays are collected in the focus of parallel rays. 

PROP. XX. When the radiant point is at that distance from the surface, at which 
parallel rays coming through it from the other side would be collected, rays flowing 
from that point become parallel on the other side. 

Plate 6, It manifestly follows from Prop. XII. that if the parallel rays AB, ID, ST, in passing through CDE, 

Fig. 13# are brought to a focus in K, rays from K as a radiant point will, after refraction, proceed in the paral¬ 

lel lines BA, Dl, TS. 

PROP. XXL When rays pass out of one medium into another of different density 
through a plane surface, if they diverge, the distance of the imaginary radiant will be 
to that of the real radiant;—if they converge, the distance of the real focus will be 

to that of the imaginary focus of the incident rays, as the sine of the angle of inci¬ 

dence is to that of the angle of refraction. 


Chap. II 


OF REFRACTION. 


Ill 


This proposition admits of four cases. 

Case 1. Of diverging- rays passing out of a rarer into a denser medium. 

Let X represent a rarer, and Z a denser medium, separated from each other by the plane surface pj ate g 
AB; suppose CD and CE to be two diverging rays proceeding from the point C, the one perpendicu- Fig. 16. 
lar to the surface, the other oblique ; through E draw the perpendicular PK. The ray CD being 
perpendicular to the surface, will proceed on in the right line CQ, but the other falling upon it ob¬ 
liquely at E, and there entering a denser medium, will suffer a refraction toward the perpendicular 
EK. Let then EG be the refracted ray, and produce it back till it intersects DC produced also in F ; 
this will be the imaginary radiant. On the centre E with the radius EF, describe the circle AFBQ, 
and produce EC to H; draw HI the sine of the angle of incidence, and GK that of refraction; equal 
to this is FP or CM, which let be drawn. Now if we suppose the points D and E contiguous, or near¬ 
ly so, then will the line HE be almost coincident with FD, and therefore FD will be to CD, as HE to 
CE or (El. VI. 4.) as HI to CM; that is, the distance of the imaginary radiant of the ray EG, is to the 
distance of the real radiant, as the sine of the angle of incidence is to that of the angle of refraction. * 

Case 2. Of diverging rays proceeding out of a denser into a rarer medium. 

Let X be the denser, Z the rarer medium, FD and FE two diverging rays proceeding from the point 
F ; and supposingjfie perpendicular PK drawn as before, FP will be the sine of the angle of incidence of 
the oblique ray FE, which in this case being refracted from the perpendicular, will pass on in some 
line as ER, which being produced back to the circumference of the circle will cut the ray FD some¬ 
where, suppose in C ; this therefore will be the imaginary focus of the refracted ray ER; draw RO, 
the sine of the angle of refraction, to which HI will be equal; but here also FP or its equal CM, is to 
HI, as EC to EH, or (if the points D and E be considered as contiguous) as DC to DF; that is, the sine 
of the angle of incidence is to the sine of the angle of refraction, as the distance of the imaginary radiant 
to that of the real radiant. 

Con. The image R of a small object G, placed under water, is one fourth nearer the surface than the 
object. And hence the bottom of a pond, river, &,c. is one third deeper than it appears to a spectator. 

If a river appear to be only 4 \ feet deep, it will be 6 feet ; a person not apprised of this, might 
venture into the water at the hazard of his life. And hence also we have the reason of the common 
phenomenon of a shilling, or other object, placed in an empty vessel, appearing to be elevated higher 
and higher as the vessel is filled with water. Suppose the vessel empty, CQ, its side opaque, G the object; 
if the eye be at H, the object will be hidden by the side CQ; but by filling the vessel with water up 
to DB, it will become visible, and seen at R, the ray EG being refracted into HE. And if the eye be 
so placed as to see the object at G when the vessel was empty, while it is filling the object will appear 
to rise gradually. Hence appears the reason why a straight stick HER, (Fig. 16.) immersed obliquely 
into water, appears bent at the surface DB ; the image of the part within lying above the object. 

Case 3. Of converging rays passing out of a denser medium into a rarer. 

Next; let Z be the denser, X the rarer medium, and GE the incident ray ; this will be refracted p]ate 6 
from the perpendicular into a line as EH; then all things x'emaining as before, GK, or its equal FP, Fig. 16. 
or CM, will be the sine of the angle of incidence, and HI that of refraction ; but these lines, as before, 
are to each other, as DC to DF; that is, the focal distance is to the distance of the imaginary focus, 
as the sine of the angle of incidence to that of the angle of refraction. 

Case 4. Of converging rays passing out of a rarer into a denser medium. 

Let Z be the rarer, X the denser medium, and RE the incident ray ; this will be refracted toward 
the perpendicular into a line, as EF; C will be the imaginary focus, and F the real one ; HI, which 
(El. I. 26.) is equal to RO, will be the sine of the angle of incidence, and FP that of the angle of re- 

* Whereas IE is to ME, or NDto CD, as HI to CM, that is, as the sine of the. angle of incidence to that of the angle of 
refraction, which (Prop. Xlli ) is always the same, the line IN is in all inclinations of the ray CE, at the same distance 
from CM. Consequently, had CE been coincident with CD, the point H had fallen upon N; and because the circle 
passes through both H and F, F would also have fallen upon N; upon which account the imaginary radiant of the ray EG 
would have been there. But the ray CE being oblique to the surface DB, the point H is at some distance from N ; and 
therefore the point F is necessarily so too, and the more so, the greater that distance is. Hence it is manifest, that no two 
rays flowing from the radiant point C, and falling with diff'ereut obliquities on the surface BD, will, after refraction there, 
proceed as from the same point; therefore, strictly speaking, there is no one point in the line D produced, that can more 
properly be called the focus of rays flowing from C, than another; for those which enter the refracting surface near D, 
will after refraction proceed, as has been observed, from the parts about N ; those which enter near E, will flow as from 
the parts about F ; those which enter about T, as from some points in the line DF produced, Sic. And it is farther to be 
observed, that when the angle DCE becomes large, the line NF increases apace; whence those rays which fall near T proceed 
after refraction, as from a more diffused space, than those which fall at the same distance from each other near the point D. 

Upon which account it is usual, with optical writers, to suppose the distance between the points where the rays enter the 
plane surface of a refracting medium, to be inconsiderable with regard to the distance of the radiant point, if they diverge ; 
or to that of their imaginary focus, if they converge ; and unless there be some particular reason to the contrary, they 
consider them as entering the refracting medium in a direction as nearly perpendicular to its surfaces as may be. 


112 


OF OPTICS. 


Book VI. 


Plate 6. 
Fig. 17. 


Plate 6. 
Fig. 17. 


t 


fraction; but these are to each other (El. VI. 2.) as DF to DC; and therefore the focal distance, is to 
that of the imaginary focus, as the sine of the angle of incidence is to that of the angle of refraction. 

PROP. XXII. When parallel rays fall upon a spheric surface of different den¬ 
sity, the focal distance will be to the distance of the centre of convexity, as the sine 
of the angle of incidence is to. the difference between that sine and the sine of the 
angle of refraction. 

Case 1. Of parallel rays passing oilt of a rarer into a denser medium through a convex surface of 
the denser. 

Let AB represent a convex surface ; C its centre of convexity; HA and DB two parallel rays, pass¬ 
ing out of a rarer medium into a denser, the one perpendicular to the refracting surface, the other 
oblique ; draw CB ; this being a radius, will be perpendicular to the surface at the point B ; and the 
oblique ray DB being in this case refracted toward the perpendicular, will proceed in some line, 
as BF, meeting the other ray in F, which will therefore be the focal point; produce CB to N, then 
will DBN, or (El. I. 20.) its equal BCA b^ the angle of incidence, and FBC that of refraction. Now, 
since any angle has the same sine with its supplement to two right ones, the angle of FCB being the 
supplement of ACB, which is equal to the angle of incidence, may here be taken for that angle ; and 
therefore, as the sides of a triangle have the same relation to each other as the sines of their opposite 
angles have, FB being opposite to this angle, and FC being opposite to the angle of refraction, they 
may here be considered as the sines of the angles of incidence and of refraction; and for the same reason 
CB may be considered as the sine of the angle CFB, which angle being, together with the angle FBC, 
equal to the external one ACB (El. I. 32.) it is itself equal to the difference between those two last angles ; 
and therefore the line FB is to CB, as the sine of the angle of incidence is to the sine of an angle which 
is equal to the difference between the angle of incidence and of refraction. Now because in very 
small angles as these are, for we suppose in this case also the distance AB to vanish (the reason of which 
will be shown in the note), their sines will have nearly the same ratio to each other that they them¬ 
selves have, the distance FB will be to CB as the sine of the angle of incidence is to the difference 
between that sine and the sine of the angle of refraction ; but because BA vanishes, FB and FA are 
equal, and therefore FA is to CA in that ratio. * 

Case 2. Of parallel ray's passing out of a denser into a rarer medium through a concave surface of 
the denser. 

Let AB be the concave surface of the denser medium, C the centre of convexity, and HA and DB two 
parallel rays. Through B, the point where the oblique ray DB enters the rarer medium, draw the 
perpendicular CN ; and let the ray DB, being in this case refracted from the perpendicular, proceed in the 
direction BM; produce BM back to H ; this will be the imaginary radiant, and DBN, or its equal ACB, 
will be the angle of incidence, and CBM, or its equal (El. I. 15.) HBN, that of refraction; then because 
NBD and DBH together are equal to NBH, the angle of refraction, therefore BCA, which is equal to 
the first, and AHB, which is equal to the second, are together equal to the angle of refraction ; and there¬ 
fore, since one of them, BCA, is equal to the angle of incidence, the other, AHB, is the difference between 
that angle and the angle of refraction. Now the sine of the angle BCA, the angle of incidence, is to the 
sine of the angle AHB, as BH to BC ; but the distance AB vanishing, HB is to CB, as HA to CA ; that is, 
the sine of the angle of incidence is to the sine of an angle which is the difference between the sine of 
the angle of incidence and that of refraction, as the distance of the focus from the surface is to that of 
the centre from the same. 

Case 3. Of parallel rays passing out of a rarer into a denser medium through a concave surface of 
the denser. 

Let AB be the concave surface of the denser medium, and let LB and FA be the incident rays. 
Now whereas, when DB was the incident ray, and passed out of a rarer into a denser medium, as in Case 
the first, it was refracted into aline BF, this ray LB having the same inclination to the perpendicular, 
will also suffer the same degree of refraction, and will therefore pass on afterward in the line FB pro- 


* It appears from the above proposition, that the focal distance of the oblique ray DB is such, that the line BF shall be 
lo the line CB or CA as the sine of the angle of incidence to the sine of an angle equal to the difference between the angle 
of incidence and of refraction ; therefore so long as the angles BCA, kc. are small, so long the line FB is nearly of the same 
length, because small angles have nearly the same ratio to each other that their sines have. But when the point B is re¬ 
moved far from A, so that the ray DB enters the surface, suppose about O, the angles BCA, &,c. becoming large, the sine of the 
angle of incidence begins to bear a considerably less ratio to the sine of an angle which is equal to the difference between 
the angle of incidence and of refraction than before, and therefore the line BF begins to bear a much less ratio to BC ; where¬ 
fore its length decreases apace ; upon which account those rays which enter the surface about 0, not only meet nearer 
the centre of convexity than those w hich enter at A, but are collected into a more diffused space. Hence it is, that the 
point where those only which enter near A, are collected, is reckoned the true focus; and the distance AB, in all demon¬ 
strations relating to the foci of parallel rays entering a spherical surface, whether convex or concave, is supposed to vanish. 


Chap. II. 


OF REFRACTION. 


113 


ducqd, toward P. So that, whereas in that case the point F was the real focus of the incident ray DB, 
the same point will, in this case, be the imaginary radiant of the incident ray LB ; but it was there 
demonstrated, that the distance FA is to CA, as the sine of the angle of incidence is to the difference be¬ 
tween that and the sine of the angle of refraction, therefore the radiant distance of the refracted ray 
BP is to the distance of the centre of convexity in that ratio. 

Case 4. Of parallel rays passing out of a denser into a rarer medium through a convex surface of the 
denser. 

Let AB be the convex surface of the denser medium, and let LB and FA be the incident rays, as be¬ 
fore. Now whereas, when DB was the incident ray passing out of a denser into a rarer medium, it was 
refracted into BM, as in Case the second, having a point as H in the line MB produced for its imaginary 
radiant ; therefore LB, for the like reason as was given in the last case, will in this be refracted into 
BH, having the same point LI for its real focus. So that here also the focal distance will be to that of 
the centre of convexity, as the sine of the angle of incidence is to the difference between that and the sine, 
of the angle of refraction. 

Cor. 1. Hence, the sines of the angles of incidence and of refraction of parallel rays being given, 
and also the distance of the centre of convexity from the surface, the focus of any lens may be easily 
found. 

Cor. 2. The distance of the centre of a glass sphere, from the principal focus of rays refracted by it, 
is nearly equal to a radius and a half of the sphere. For in this case it will be FA : CA :: 1: 21—2R, 

3CA 3C 4 

(where I and R signify the sines of incidence and refraction’); therefore FA=-—And if the 

4CA °—4 2 

sphere be of water instead of glass, FA=- T y—=2CA= the diameter of the sphere. 

PROP. XXIII. When diverging or converging rays enter into a medium of dif¬ 
ferent density through a spherical surface, the ratio compounded of that which the focal 
distance bears to the distance of the radiant point (or of the imaginary focus of the 
incident rays, if they converge) and of that, which the distance between the same 
radiant point (or imaginary focus) and the centre, bears to the distance between the 
centre and the focus, is equal to the ratio, which the sine of the angle of incidence bears 
to the sine of the angle of refraction ; that is, FD x CA : DA x CF :: FB : BG. 

Case 1. Of diverging rays passing out of a rarer into a denser medium through a convex surface of 
the denser, with such a degree of divergency, that they shall converge after refraction. 

Let BD represent a spherical surface, and C its centre of convexity ; and let there be two diverging p ]ate 6 
rays, AB and AD, proceeding from the radiant point A, the one perpendicular to the surface, the other Fig. 19.* 
oblique. Through the centre C produce the perpendicular ray AD to F, and draw the radius CB and 
produce it to K, and let BF be the refracted ray; then will F be the focal point; produce AB to H, and 
through the point F draw the line FG parallel to CB. AB being the incident ray, and CK perpendicular 
to the surface at the point B, the angle ABK, or which is equal to it, because of the parallel lines CB 
and FG, FGH is the angle of incidence. Now since the supplement of any angle to two right ones has 
the same sine with the angle itself, the sine of the angle FGB, which is the supplement of FGH to two 
right ones, may be considered as the sine of the angle of incidence ; for which sine the line FB, as the 
sides of a triangle have the same ratio to each other, that the sines of their opposite angles have, may 
be taken. Again, the angle FBC is the angle of refraction, or its equal, because alternate to it, 

BFG, to which BG being an opposite side, may be taken as the sine. But FB is to BG in a ratio com¬ 
pounded of FB to BA, and of BA to BG; for the ratio that any two quantities bear to each other is 
compounded of the ratio which the first bears to any other, and of the ratio which that other bears 
to the second. Now FB is to BA, supposing BD to vanish, as FD to DA ; and BA is to BG, because of 
the parallel lines CB and FG, as AC to CF. That is, the ratio compounded of FD, the focal distance, to 
DA, the distance of the radiant point, and of AC, the distance between the radiant point and the centre, 
to CF, the distance between the centre and the focus, is equal to that which the sine of the angle of 
incidence bears to the sine of the angle of refraction.* 

* Since the focal distance of the oblique ray AB is such that the compound ratio ofFB to BA and of AC to CF shall be 
the same, whatever be the distance between B and D ; it is evident that, AC being always of the same length, the more the 
line AB lengthens, the more FB must lengthen too, or else FC must shorten ; but if BF lengthens, CF will do so too, and in 
a greater ratio with respect to its own length than BF will, therefore the lengthening of BF will conduce nothing toward 
preserving the equality of the ratio ; but as AB lengthens, BF and CF must both shorten, which is the only possible way in 
which the ratio can be continued the same. And it is also apparent, that the farther B moves from D toward O, the faster 

15 


114 


Plate 6. 
Fig. 19. 


Plate 6. 
Fig. 18. 


Plate 6. 
Fig. 20. 


OF OPTICS. Book VI. 

Case 2. Of converging rays passing out of a rarer into a denser medium through a concave surface 
of the denser with such a degree of convergency, that they shall diverge after refraction. 

Let the incident rays be HB and FD passing out of a rarer into a denser medium through the con¬ 
cave surface BD, and tending toward the point A, from whence the diverging rays flowed in the other 
case ; then the oblique ray HB having its angle of incidence HBC equal to ABK, the angle of incidence 
in the former case, will be refracted into the line BL, such that its angle of refraction KBL will be 
equal to FBC, the angle of refraction in the former case ; that is, it will proceed after refraction in the 
line FB produced, having the distance of the imaginary radiant the same as the focal distance FD with 
the diverging rays AB, AD, in the other case. But, by what has been already demonstrated, the ratio 
compounded of FD, the distance of the imaginary radiant, to DA, in this case, the distance of the imagi¬ 
nary focus of the incident rays, and of AC, the distance between the same imaginary focus and the centre, 
to CF, the distance between the centre and the imaginary radiant, is equal to that which the sine of the 
angle of incidence bears to the sine of the angle l of refraction. 

Case 3. Of diverging rays passing out of a rarer into a denser medium through a convex surface of 
the denser, with such a degree of divergency as to continue diverging. 

Let AB, AD, be the diverging rays, and let their divergency be so great, that the refracted ray BL 
shall also diverge from the other; produce LB back to F, which will be the imaginary radiant; draw 
the radius CB, and produce it to K ; produce BA likewise toward G, and draw FG parallel toBC. Then 
will ABK be the angle of incidence, whose sine BF may be taken for, as being opposite to the angle 
BGF, which is the supplement of the other to two right ones. And LBC is the angle of refraction, or 
its equal KBF, or which is equal to this, BFG, as being alternate ; therefore BG, the opposite side to 
this may be taken for the sine of the angle of refraction. But BF is to BG, for the like reason as was 
given in Case the first, in a ratio compounded of BF to BA, and of BA to BG. Now BF is to BA, (DB 
vanishing) as DF to DA, and because of the parallel lines FG and BC, the triangles CBA and AGF are 
similar, therefore BA is to AG, as CA to AF; consequently, BA is to BA together with AG, that is, to 
BG, as CA is to CA together with AF, that is, CF. Therefore the ratio compounded of DF, the distance 
of the imaginary radiant, to DA, the distance of the real radiant point, and of CA, the distance between 
the real radiant point and the centre, to CF, the distance between the centre and the imaginary radiant, 
is equal to that which the sine of the angle of incidence bears to the sine of the angle of refraction. 

Schol. By making HB and CD the incident rays, the proposition may be proved of converging rays 
passing out of a rarer into a denser medium, through a concave surface of a denser, with such conver- 
gencjr, that they shall continue to converge. Also, by the same method of reasoning as in the preceding 
cases, the proposition may be proved, of diverging rays passing out of a denser into a rarer medium 
through a concave surface of the denser, and of converging rays passing out of a denser into a rarer me¬ 
dium through a convex surface of the denser. And all those cases which are the converse of the pre¬ 
ceding, admit of a similar proof; for, when rays pass out of one medium into another, the sine of the 
angle of incidence has the same ratio to the sine of the angle of refraction, as the sine of the angle of 
refraction has to the sine of the angle of incidence, when they pass through the same lines of direction 
the contrary way. 

Case 4. Of rays passing out of a denser into a rarer medium, from a point between the centre of 
convexity and the surface. 

Let AB, AD, be two rays passing out of a denser into a rarer medium, from the point A, which is 
taken between C, the centre of convexity, and the refracting surface BD; through B draw CK, and let 
BL be the refracted ray; produce BL back to F, and draw FG parallel to BC. Then will ABC be the 
angle of incidence, of which BF, being opposite to its alternate and equal angle BGF, is the sine. LBK 
will be the angle of refraction, or its equal FBC, of which BG, being opposite to its supplement to two 
right angles BFG, is the sine. But, BF is to BG in the compound ratio of BF to BA, and of BA to BG; 
and (BD vanishing) BF is to BA as DF to DA, and because the lines CB and FG are parallel, BA is to 
BG as CA to CF. 

Case 5. Of rays passing out of a rarer into a denser medium from a point between the centre of 
concavity and the surface. 

AB lengthens, and therefore the farther the rays enter from D, the nearer to the refracting surface is the place where they 
meet, but the space they are collected in is the more diffused ; and therefore in this case, as well as those taken notice of in 
the two preceding notes, different rays, though flowing from the same point, will constitute different foci; and none are 
so effectual as those which enter at or very near the point D. And since the same is observable of converging as well as of 
diverging rays, none, except those which enter very near that point, are usually taken into consideration ; upon which ac¬ 
count it is, that the distance DB, in determining the focal distances of diverging or converging rays entering a convex or 
concave surface, is supposed to vanish. 


Chap. II. 


OF REFRACTION. 


11 


Let AB, AD, be two diverging rays passing out of a rarer into a denser medium through the refract- p] a ( e e. 
ing surface BD, whose centre of concavity is C, a point bejmnd that from whence the rays flow. Fig. 21. 
Through B draw CK, and let BL be the refracted ray; produce it back to F, and draw FG parallel to 
BC, meeting BA in G. ABC will be the angle of incidence, of which BF, being opposite to its supple¬ 
ment to two right angles BGF, is the sine. The angle of refraction is LBK, or its equal FBC, of which 
BG, being opposite to its alternate and equal angle BFG, is the sine. But BF is to BG in the compound 
ratio of BF to BA and of BA to BG; and (BD vanishing) BF is to BA as DF to DA; and because of the 
parallel lines CB and GF, the triangles AFG and ABC are similar. BA therefore is to AG, as CA to 
AF; consequently, BA is to BA — AG, that is, to BG, as CA is to CA — AF, that is, to CF. 

In like manner the Proposition may be proved of rays passing out of a denser into a rarer medium 
toward a point between the centre of convexity and the surface, and in all other supposable cases. 

Cor. Hence the distance of the radiant point, or of the imaginary focus, being given, and also the 
radius of convexity, and the sines of the angles of incidence and refraction in the two mediums, the 
focus of any lens may be thus found. 

Let it be required to determine the focal distance of diverging rays passing out of air into glass through 
a convex surface, and let the distance of the radiant point be 20, and the radius of convexity be 5 ; let 
the focal distance be expressed by x ; then, because by the preceding Proposition the ratio compounded 
of that which the focal distance bears to the distance of the radiant point, (that is, in this supposition, ot 
x to 20,) and of the ratio which the distance of the same radiant point from the centre bears to the dis¬ 
tance between the centre and the focus (in this case, of 25 to x — 5,) is equal to the ratio which the 
sine of the angle of incidence bears to the sine of the angle of refraction (that is, of 17 to 11), wre shall 

have in the instance before us the following proportion, f 20 _ ^ : 17 : 11, and compounding them 

in one, which is done by multiplying the two first parts together, we have 25x : 20x 100 :: 17 : 11, 

v _ 1700 

x — 65 • 


SECT. II. 

Of Images produced bij Refraction. 

PROP. XXIV. Rays of light flowing from the several points of any object, farther 
from a convex lens than its principal focus, by passing through the lens, will be made 
to converge to points corresponding to those from which they proceeded, and will form 
an image. 

Let ABC be a luminous or illuminated object. From every point, as A, B, C, rays diverge in all di- pj ate 6 
rections. Let some of these rays fall upon a convex lens GHK placed in a hole GK, in the window Fig. 22. 
shutter of a dark room ML, at a greater distance from the object than the principal focal distance of the 
lens. BH being the axis, will (by Prop. XIV.) pass through the lens without refraction in the direction 
BHE. But the collateral rays BG, BK, made equally convergent by the lens, will cross the axis at E ; 
that is, all the rays which come from the point B in the object, will be united behind the lens in the 
focus E. In like manner, among the rays AG, AH, AK, which diverge from the point A, whilst AH the 
axis (as was shown Prop. XVII. Schol.) may be considered as if it went straight through the lens, the 
other rays will be made to converge, and will be united in a focus at F; and also, the rays from C will 
be united in D. The same may be shown concerning every other point in the object. Consequently, 
there will be as many correspondent foci in the image as there are radiant points in the object; and 
these foci will be disposed in the same manner with respect to one another as the radiants, and will 
therefore form an image. The object must be farther from the lens than its principal focus, else the 
rays from the several radiants would not converge, but either become parallel or diverging (by Prop. 

XVIL), whence no image would be formed. 

Exr.-l. Let the rays of the sun pass through a convex lens into a dark room, and fall upon a sheet of 
white paper placed at the distance of the principal focus from the lens. 

2. The rays of a candle, in a room from which all external light is excluded, passing through a con¬ 
vex lens, will form an image on w r hite paper. 

PROP. XXV. The image produced by rays of light passing through a convex lens 
is inverted. 


Book VI. 


116 


Plate 6. 
Fig. 22. 


Plate 6. 
Fig. 22. 


Plate 6. 
Fig. 22. 


OF OPTICS. 

The focus in which the rays, that come from any point A, or B, are united, is in the axis AHF, or 
BHE, of the beam, whether it fall directly or obliquely upon the lens. But the axis (by Prop. XVII.) 
is the middle ray of a cone of rays whose base is the surface of the lens, and vertex the radiant point. 
Every axis, therefore, as AHF, BHE, must pass through H, the middle point of the lens, and conse¬ 
quently must cross one another in that point; from which it is manifest, that the rays from the lowest 
point C of the object will become the highest point of the image D; and that the image will be, with 
respect to the original object, inverted. 

PROP. XXVI. The image will not be distinct, unless the plane surface, on which 
it is received, be placed at the distance of the principal focus of the lens. 

For otherwise the rays which come from a single point in the object, will not have its correspond¬ 
ing point in the image, but will be spread over a larger surface. 

PROP. XXVII. As an object approaches a convex lens, its image departs from it, 
and as the object departs, the image approaches. 

As the object ABC approaches the lens, the several radiants approach it; and consequently (by 
Prop. XIX.) the several foci which form the image FED recede; and the reverse. But the image can 
never be nearer the lens than its focus of parallel rays, since this is the place of the image, when the 
object is infinitely distant. 

PROP. XXVIII. When the object is placed parallel to the image, the diameter of 
the object is to the diameter of the distinct image, as the distance of the object from 
the lens, is to the distance of the image from the lens. 

The radiant A (as appears from Prop. XXV.) is represented by its focus in the point F, where the 
line AH, produced behind the lens, cuts a plane passing through the focus of parallel rays parallel to the 
lens. In like manner the radiant C is represented by its corresponding focus in the point D, in which 
the same plane is cut by the line CH produced. If therefore the distance of the extremities of the ob¬ 
ject, or its length, be AC’, the length of its image will be DF. Since therefore AC is parallel to DF, 
the alternate angles ACD, CDF, (El. I. 29.) are equal; and also the alternate angles CAF, AFD; whence 
(El. VI. 4.) AC is to FD, as CH to DH, as BH is to EH, that is, the height of the object is to that of 
the image, as the distance of the object from the lens, to the distance of the image from the lens. Any 
diameter or line drawn across the object may be proved, in like manner, to have the same ratio to any 
corresponding diameter or line drawn across the image. 

PROP. XXIX. When the image appears confused, it is larger than when it is 
distinct. 

For the rays, in this case, are not received upon the white surface exactly at the distance from the 
lens at which they are brought to a focal point, but either at a distance greater or less ; and in either 
case the rays which come from any radiant points at the extremities A and C, will not be collected into 
points on the plane at F and D, but be spread over a small circular space round these points ; whence 
the confused image will be larger than the distinct image. 

PROP. XXX. The object and distinct image are similar surfaces. 

Though the side of any object which is toward the lens be not a plane surface, yet the light is re¬ 
flected from it in the same manner as if the figure of the object were drawn upon the plane surface of a 
piece of canvass, and differently shaded. Therefore the side of the object next to the lens may be con¬ 
sidered as a plane figure. And since (by Prop. XXVIII.) the height of the object is to that of the pic¬ 
ture, as the distance of the object from the lens, to the distance of the image from the lens, and also the 
breadth of the object in any part, to the breadth of the image in the corresponding part, in the ratio of 
these distances; it follows (El. V. 11.) that the height of the object is to the height of the image, as 
the breadth of the object in any part is to the breadth of the image in its corresponding part; that is, 
The object and image are similar surfaces. 

PROP. XXXI. The diameter of an image formed by rays passing from a given 
object through a convex lens, increases as the object approaches the lens, and de¬ 
creases as the object recedes from the lens. 

The diameter of the image (by Prop. XXVIII.) increases as its distance increases, and decreases 


Chap. II. 


OF REFRACTION. 


117 


as its distance decreases. And (by Prop. XXXVII.) the distance of the image increases as the distance 
of the object decreases, and the reverse. Whence the diameter of the image increases as the distance 
of the object decreases, and decreases as the distance of the object increases. 

PROP. XXXII. When the distance of the object is given, the diameter of the 
image is as the diameter of the object. 

If the object AC remain at the distance BH from the lens, the image (by Prop. XXVII.) will re- Plate 6. 
main at the distance EH ; whence the ratio of BH to EH, and consequently (by Prop. XXVIII.) the Fig. 22. 
ratio of the diameter AC to its correspondent diameter DF, is given, or is invariable. Consequently, 
if AC increases or decreases, DF must proportionally increase or decrease ; that is, the diameter of 
the image is directly as the diameter of the object. 

PROP. XXXIII. When the diameter and distance of the object are given, the 
diameter of the image will be as its distance from the lens. 

If the diameter and distance of the object are given, it is manifest that the diameter of the image 
Gannot be varied without changing the lens. But if, instead of the lens GHK, one less convex, or more 
convex, be used, the rays will be brought to a focus, and the image (by Prop. XXIV.) will be formed 
at a greater or less distance from the lens. And since (by Prop. XXVIII). the distance of the object 
is always to the distance of the image, as the diameter of the object to the diameter of the image ; 
the first and third terms remaining invariable, the second and fourth must be increased or diminish¬ 
ed proportionally; that is, the diameter of an image will be directly as its distance from the lens. 

PROP. XXXIV. When the diameter and distance of the object are given, the 
area of the image is as the square of its distance from the lens. 

Because the surface of the image (by Prop. XXX.) is similar to the surface of the object, whilst 
the surface of the object remains the same, the image, in every variation of its magnitude, must be 
similar to itself. But the areas of similar surfaces (El. VI. 20. Cor. 1.) are as the squares of their 
homologous sides, that is, as the squares of their heights or breadths. Therefore the area of the im¬ 
age, is always as the square of its diameter. And the diameter of the image, when the diameter and 
distance of the object are given, is (by Prop XXXIII.) as its distance from the lens; therefore the 
square of its diameter, or its area, is as the square of its distance from the lens. 

PROP. XXXV. Though the distance of the object from the lens be varied, the 
image may be preserved distinct without varying the distance of the plane surface 
which receives it. 

This will be the case, if as much as the image is brought forward by the removal of the object, 
it is thrown backward by diminishing the convexity of the lens, and the reverse ; or the image may be 
preserved distinct without changing the lens, by increasing or diminishing the distance of the lens from 
the plane surface which receives the image, in the same ratio as the distance of the object from the 
lens is increased or diminished; w’hich may be done either by moving the lens or the plane surface. 

PROP. XXXVI. The distances of the object and image, and the diameter of the 
object being given, the diameter of the image will not be altered by changing the 
area of the lens. 

The height of the image DF is the distance between the two extreme’ foci F and D ; the former Plate 6 . 
of which is always in the axis AHF of the cone which has A for its vertex, and the latter in the axis Fig. 22. 
CHD of the cone whose vertex is C, which axes cross each other in H. Since therefore DF, the 
the height of the image, is the distance betw-een these two lines AIIF, CHD, where they meet the 
plane surface, the height of the image will be the same, whether the whole area, GHK is open, or 
only a small part of it at H. 

PROP. XXXVII. When the object is near the lens, but not so near as the prin¬ 
cipal focus, in order to make the image distinct, the area of the lens must be small. 

If the object was as near to the lens as the principal focus, or nearer, no image (by Prop. XVII.) Plate 6. 
could be formed. But let the object A be at, a distance from the lens NP, very little greater than Fig. 23. 
that of the principal focus; then the extreme rays AN, AP, of the cone NAP, diverging more than 
the rays AD, AE, if the plane surface, which is to receive the rays, is placed where these rays are col¬ 
lected to a focus, the extreme rays AN, AP, diverging more, will not be collected, and the image on 


118 


OF OPTICS. 


Book VI. 


Plate 6. 
Fig. 22. 


Plate 12. 
Fig. 9. 


the plane surface will be confused. To prevent this, the extreme rays must be excluded by dimin¬ 
ishing the area of the lens, or of the hole where it is placed. If the radiant A were at a greater 
distance, this would be unnecessary. Supposing the lens at SR, the extreme rays AN, AP, would pass 
above or below the lens, and only the middle rays, .which are brought to a focus on the plane surface, 
would pass through the lens. 

PROP. XXXVIII. The brightness of an image, when its distance from the lens 
is given, is as the area of the lens. 

When the whole area GHK is open, the entire cone of rays AGK passes through the lens from 
the point A, and is brought to a focus at F ; but when the area is diminished to a small surface at H, the 
greatest part of the cone is excluded, and no rays, but the axis AH and those which are near it can 
pass through the lens ; whence it is manifest, that the focal point F must be more illuminated by the 
rays from A when the area of the lens is GHK, than when the area is diminished. The same may 
be said of every other cone of rays, and of every other point in the image. Therefore the whole 
image, although (by Prop. XXXVII.) made more distinct by diminishing the area, will be made faint¬ 
er or less bright. 

PROP. XXXIX. The brightness of the image, when the area of the lens and the 
distance of the object are given, is inversely as the square of its distance from the 
lens. 

The area of the lens and distance of the object being given, the number of rays which pass 
through the lens and form the image, is given. Now the same number of rays spread over a larger 
surface will not illuminate it so strong]}'- as they would a smaller surface ; that is, the brightness 
will be inversely as the illuminated area; and the area of the image is (by Prop. XXXIV.) as the 
square of its distance from the lens; whence its brightness is inversely as the square of its distance. 

PROP. XL. The heat at the focus of a burning glass, when the area of the glass 
is given, is inversely as the square of the focal distance. 

The distance of the burning spot, that is, the image of the sun, from the lens, is the focal dist¬ 
ance, because the sun’s rays are parallel. And because the heat and the brightness at the focus are as 
the number of rays collected, the heat is as the brightness. But the brightness (by Prop. XXXIX.) is 
inversely as the square of the distance of the image from the lens; therefore the heat is in the same 
ratio, that is, in this case, inversely as the square of the focal distance of the glass. 

PROP. XLI. The heat at the focus of a burning glass, when the focal distance is 
given, is as the area of the glass. 

The brightness is (by Prop. XXXVIII.) as the area of the lens, and the heat is as the brightness; 
therefore the heat is also as the area of the lens. 

PROP. XLII. The heat at the focus of a burning glass is to the common heat of 
the sun, inversely as the area of the focus to the area of the glass. 

The brightness, or the heat, must be inversely as the space or area over which the rays which 
cause them are spread, that is, inversely as the area of the focus to the area of the glass. 

Schol. This proposition supposes all the rays which fall upon the lens to pass through it to the 
focus. 

Exp. Most of the preceding Propositions from Prop. XIX. to XXXIX. may be confirmed, in a room 
from which all external light is excluded, by placing a convex lens x, fixed in a frame which moves 
perpendicularly upon an oblong bar of wood, or table BD, at distances such as the Propositions re¬ 
quire from a lighted candle Q, placed perpendicularly on the same bar of wood, and receiving the im¬ 
ages upon white paper q. Upon this bar of wood, on one side of a line over which the convex 
lens is placed, let a line perpendicular to the last mentioned line be divided into parts 1, 2,3,4, 
&c. each equal to the distance of the focus of parallel rays; and on the other side of the lens 
let a line be divided in the same manner, and let the first division which is farther from the lens than 
the focus, be subdivided into parts respectively equal to i, i, 1, Sic. of the distance of the focus of 
parallel rays; if a candle be placed over the division 2, it will form a distinct image on a paper held 
over the division F; if the candle be over 3, the image will be at A, &ic. whence it appears, that the 
distances of the correspondent foci vary reciprocally. Prop. XL. XLI. XLII. may be confirmed by hold¬ 
ing a large double convex lens, or burning glass, in the sun’s rays, and receiving the image on white 
paper, or other substances. 


Chap. III. 


OF REFLECTION. 


119 


CHAPTER. III. 

Of Reflection. 

SECTION I. 

OF THE LAWS OF REFLECTION. 

Def. XIX. A Ray of Light, turned back into the same medium in which it mov¬ 
ed before its return, is said to be reflected. 

Def. XX. The Angle of reflection is that, which is contained between the line 
described by a reflected ray and a line perpendicular to the reflecting surface at the 
point of reflection. 

Let AC be the incident ray falling upon the reflecting surface DE, CB will be the reflected ray, piate 6. 
OC the perpendicular, ACO the angle of incidence, and OCB the angle of reflection. Fig. 25 

PROP. XLIII. The reflection of light from transparent bodies is either partial or 
total; the partial reflection happens either at the first or second surface, the total, 
at the second surface only. 

When a beam of light falls upon a thick piece of polished glass, all the rays will not pass through 
it • but some of them will be reflected from the first surface of the glass, where the beam enters. 

At the second surface, some of the rays will also be l’eflected. These partial reflections happen, 
whatever is the obliquity of the rays. The total reflection happens when the angle of incidence, or 
the obliquity, is greater than 41 degrees. All the light which then comes to the second surface will 
be reflected.' See Cor. 3. Prop. XIII. 

Schol. The rays of light are not reflected by striking upon the solid parts of bodies. 

At least as many rays are reflected from the second surface when the light passes out of glass 
into air, as from the first when it passes out of air into glass; but if the reflection were caused by 
the striking of the rays upon solid parts of bodies, since glass is denser than air, that is, has more solid 
parts in a given space, a greater quantity of rays would be reflected from the first surface than from 
the second. Besides, it seems improbable that, at the second surface, with one degree of obliquity, the 
rays should meet with nothing but pores or interstices in the air, and pass freely into it, and that with 
a* treater degree of obliquity, it should meet with nothing but solid parts, and be totally reflected. 

Again since water is denser than air, if the reflection were owing to the striking of the rays upon the 
solid body, it might be expected that the light would be more perfectly reflected in passing out of 
glass into water than into air; whereas, it is found, that if water be placed behind the glass instead 
of air. rays will not be reflected at the second surface, though their obliquity be greater than 41 de¬ 
grees; hence also it is manifest, that the reflection is not owing to the striking of the rays upon the 
second surface of the glass; for then it would be the same, whatever were the medium beyond it. 

PROP. XLIV. Reflection is caused by the powers of attraction and repulsion in 
the reflecting bodies. 

Supposing that bodies attract those rays which are very near them, and repel those a little farther Plate 6. 
from them ; and calling the space contiguous to the surface of the glass, where the rays are attracted, F| g> 24 - 
the attracting surface; and the space next to this, the repelling surface ; the proposition may be thus 
proved. 

Let GG, MM, be the repelling surface of a piece of glass, and R« a ray falling upon it. This ray 
when it enters the surface will be retarded by the repulsion, and consequent^, refracted from the per¬ 
pendicular at a. And this repulsion increasing til! the ray gets into the surface of attraction, the ray 
will be constantly retarded, that is, turned out of its straight course at b , c, d , &c. till it becomes parallel 
to MM at/, the limit of the repelling surface. And in this situation of the ray, the repelling force, 
which had retarded, will now constantly accelerate it, and consequently it will be continually refracted 
toward the perpendicular, at g , h, i , &.c. till it emerge from the surface at l; when, the repelling force 
ceasing to act, the ray will proceed in a right line. Thus the ray, by reflection, is made to describe 
the curve af l- 


120 


Plate 6. 
Fig. 25. 


Plate 6. 
Fig. 26. 


Plate 6. 
Fig. 25. 


OF OPTICS. Book VI. 

PROP. XLV. The angle of reflection is equal to the angle of incidence, and their 
complements are also equal. 

In all cases of reflection, the rays (by Prop. XLIV.) describe such a curve at afl. And since they 
describe one half of this curve by being retarded, and one half by being similarly accelerated, one half 
will be similar to the other; whence one half will make the same angle with a perpendicular at f, as 
the other half makes with it. And the bending of the rays is made within so very small a compass, that 
is, the curve afl is so small, that it may be neglected, as in flg. 25, where the angle ACQ is equal to 
the angle BCO, and consequently, the angle ACD equal to BCE. 

Exr. 1. Having described a semicircle on a smooth board, and from the circumference let fall a per¬ 
pendicular bisecting the diameter, on each side of the perpendicular cut off equal parts of the circum¬ 
ference ; draw lines from the points in which those equal parts are cut off to the centre; place three 
pins perpendicular to the board, one at each point of section in the circumference, and one at the cen¬ 
tre ; and place the board perpendicular to a plane mirror. Then look along one of the pins in the cir¬ 
cumference to that in the centre, and the other pin in the circumference will appear in the same line 
produced with the first; which shows that the ray which comes from the second pin, is reflected from 
the mirror at the centre to the eye, in the same angle in which it fell on the mirror. 

2. Let a ray of light passing through a small hole into a dark room, be reflected from a plane mirror; 
at equal distances from the point of reflection, the incident and the reflected ray will be at the same 
height from the surface. 

PROP. XLVI. All reflection is reciprocal. 

If the ray AC, after it has been reflected in the line CB, is turned back again in the direction CB, 
it will be reflected (by Prop. XLV.) into AC. Therefore if ACO be the angle of incidence, OCB will 
be the angle of reflection ; and if OCB be the angle of incidence, ACO will be the angle of reflection. 

Schoi.. Sir Isaac Newton explains the cause of reflection by supposing, that light in its passage from 
the luminous body is disposed to be alternately reflected by, and transmitted through any refracting sur¬ 
face it may meet with ; that these dispositions which he calls Fits of easy reflection and easy transmis¬ 
sion, return successively at equal intervals; and that they are communicated to it, at its first emission 
out of the luminous body it proceeds from, probably by some very subtle and elastic substance diffused 
through the universe, in the following manner. As bodies falling into water, or passing through the air, 
cause undulations in each, so the rays of light may excite vibrations in this elastic substance. The 
quickness of these vibrations depending on the elasticity of the medium (as the quickness of the vibra¬ 
tions in the air, which propagate sound, depend solely on the elasticity of the air, and not upon the 
quickness of those in the sounding body), the motion of the particles of it may be quicker than that of 
the rays ; and therefore when a ray, at the instant it impinges upon any surface, is in that part of a vi¬ 
bration of this elastic substance which conspires with its motion, it may be easily transmitted, and when 
it is in that part of a vibration which is contrary to its motion, it may be reflected. He farther suppos¬ 
es, that when light falls upon the first surface of a body, none is reflected there, but all that happens to 
it there is, that every ray that is not in a fit of easy transmission, is there put into one, so that when 
they come at the other side (for this elastic substance easily pervading the pores of bodies, is capable 
of the same vibrations within the body as without it,) the rays of one kind shall be in a fit of easy trans¬ 
mission, and those of another in a fit of easy reflection, according to the thickness of the body, the in¬ 
tervals of the fits being different in rays of a different kind. 

PROP. XLVII. Rays of light reflected from a plane surface, have the same degree 
of inclination to each other that their respective incident ones have. 

The angles of reflection of the rays AC, AI, AK, being equal to that of their respective incident 
ones, it is evident that each reflected ray will have the same degree of inclination to the surface DE, 
from whence it is reflected, that its incident one has; but it is here supposed that all those portions 
of surface, from whence the rays are reflected, -are situated in the same plane ; consequently, the re¬ 
flected rays FC, LI, MK, will have the same degree of inclination to each other that their incident ones 
have, from whatever part of the surface they are reflected. 

Cor. Parallel rays falling on a reflecting plane surface are reflected parallel. 

PROP. B. If a plane mirror revolve upon an axis, the angular velocity of the re¬ 
flected ray is double that of the mirror. 

Let DE be a mirror, AC an incident ray, and CB a reflected ray. If the mirror turn upon an axis at 
C so as to come into the situation FG, then the incident ray AC will be reflected into the line CH. 


% 


Chap. III. 


OF REFLECTION. 


121 


Now the angle BCH, which expresses the angular velocity of the reflected ray BC, is double of the 
angle DCF, which is the angular velocity of the mirror. For the angle ACD = BCE = BCH -f* HCE. 

(Prop. XLV.) For the same reason ACF = HCG = HCE + (ECG) FCD. Therefore the angle 
DCF (ACD — ACF) = BCH — DCF, consequently the angle BCH — 2DCF; that is, the angular ve¬ 
locity of the reflected ray is double that of the mirror. 

Schol. Upon this Proposition depend the construction and use of Hadley's quadrant. 

PROP. XLVIII. Parallel rays reflected from a concave surface are made con¬ 
verging. 

Let AF, CD, EB, represent three parallel rays falling upon the concave surface FB, whose centre piaie 6. 
is C. To the points F and B draw the lines CF, CB ; these being drawn from the centre will be per- Fig. 27. 
pendicular to the surface at those points. The incident ray CD also passing through the centre will be 
perpendicular to the surface, and therefore will return after reflection in the same line; but the ob¬ 
lique rays AF and EB will be reflected into the lines FM, BM, situated on the contrary side of their re¬ 
spective perpendiculars CF and CB. They will therefore proceed converging after reflection toward 
some point, as M, in the line CD. 

PROP. XLIX. Converging rays falling upon a concave surface are made to con¬ 
verge more. 

Let GB, HF, be the incident rays. Now, because these rays have larger angles of incidence than Plate 6. 
the parallel ones, AF, EB, in the foregoing case, their angles of reflection will also be larger than Fig. 27. 
theirs; they will therefore converge after reflection, suppose in the lines FN and BN, having their 
point of concourse N farther from C than the point M, to which the parallel rays AF and EB converged 
in the foregoing case. 

PROP. L. Diverging rays, falling upon a concave surface, if they diverge from 
the focus of parallel rays, become parallel;—if from a point nearer the surface than 
that focus, will diverge less than before reflection ;—if from a point between that focus 
and the centre, will converge after reflection to some point, on the contrary side of the 
centre, and farther from the centre than the point from which they diverged ;—if from 
a point beyond the centre, the reflected rays will converge to a point on the contrary 
side, but nearer to it than the point from which they diverged ;—if from the centre, 
they will be reflected thither again. 

Let the incident diverging rays be MF, MB, proceeding from M, the focus of parallel rays; then as p] a t e 6. 
the parallel rays AF and EB were reflected into the lines FM and BM, these rays will now on the con- Fig. 27. 
trary be reflected into them. (By Prop. XLVJ.) 

Let them diverge from N, a point nearer to the surface than the focus of parallel rays; they will 
then be reflected into the diverging lines HF and BG, which the incident rays GB and HF described, 
which were shown to be reflected into them in the foregoing proposition; but the degree wherein 
they diverge, will be less than that wherein they diverged before reflection. 

Let them proceed diverging from X, a point between the focus of parallel rays and the centre; they 
then make less angles of incidence than the rays MF and MB, which became parallel by reflection; 
they will consequently have less angles of reflection, and proceed therefore converging toward some 
point, as Y ; which point will always fall on the contrary side of the centre, because a reflected ray 
always falls on the contrary side of the perpendicular with respect to that on which its incident one 
falls; and therefore will be farther distant from the centre than X. 

If the incident ones diverge from Y, they will after reflection converge to X, those which were the 
incident ra} r s in the former case being the reflected ones in this. 

Lastly, if the incident rays proceed from the centi’e, they fall in with their respective perpendicu¬ 
lars, and for that reason are reflected thither again. 

Exp. Place a concave mirror at proper distances from an open orifice,, or a convex or a concave 
lens, through which a beam of solar rays passes, according to the three preceding propositions. 

PROP. LI. Parallel rays reflected from a convex surface are made diverging. 

Let AB, GD, EF, be three parallel rays falling upon the convex surface BF, whose centre of con- Plate 6 
vexity is C, and let one of them, G’D, be perpendicular to the surface. Through B, D, and F. the Fig. 28. 

16 


122 


OF OPTICS. 


Book VI. 


Plate 6. 
Fig. 28. 


Plate 6. 
Fig. 28. 


Plate 6. 
Fig. 26. 


points of reflection, draw the lines CV, CG, and CT, which because they pass through the centre will 
be perpendicular to the surface at those points. The incident ray GD, being perpendicular to the sur¬ 
face, will return after reflection in the same line, but the oblique ones AB and EF in the lines BK and 
FL situated on the contrary side of their respective perpendiculars BV and FT. They will therefore di¬ 
verge after reflection, as from some point M in the line GD produced. 

PROP. LII. Diverging rays reflected from a convex surface are made more di¬ 
verging. 

Let GB, GF, be the incident rays. These having larger angles of incidence than the parallel ones 
AB and EF in the preceding case, iheir angles of reflection will also be larger than theirs ; they will 
therefore diverge after reflection, suppose in the lines BF and FO, as from some point N farther from 
C than the point M ; and the degree wherein they will divei’ge, will exceed that wherein they diverg¬ 
ed before reflection. 

PROP. LIII. Converging rays reflected from a convex surface, if they tend toward 
the focus of parallel rays, will become parallel;—if to a point nearer the surface than 
that focus, will converge less than before reflection;—if to a point between that 
focus and the centre, will diverge as from a point on the contrary side of the centre 
farther from it than the point toward which they converged ;—if to a point beyond 
the centre, they will diverge as from a point on the contrary side of the centre near¬ 
er to it than the point toward which they first converged;—and if toward the cen¬ 
tre, they will proceed, on reflection, as from the centre. 

Let the converging rays be KB, LF, tending toward M, the focus of parallel rays; then, as the par¬ 
allel rays AB, EF, were reflected into the lines BK and FL, those rays will now on the contrary be re¬ 
flected into them. 

Let them converge in the lines PB, OF, tending toward N, a point nearer the surface than the fo¬ 
cus of parallel rays; they will then be reflected into the converging lines BG and FG, in which the rays 
GB, GF, proceeded, w'hich were shown to be reflected into them in the proposition immediately 
foregoing ; but the degree wherein they will converge, will be less than that wherein they converged 
before reflection. 

Let them converge in the lines RB and SF proceeding toward X, a point between the focus of par¬ 
allel rays and the centre ; their angles of incidence will then be less than those of the rays KB and 
LF, which became parallel after reflection; their angles of reflection will therefore be less, on which 
account they must necessarily diverge, suppose in the lines BH and FI, from some point, as Y ; 
which point will fall on the contrary side of the centre with respect to X, and will be farther from it 
than X. 

If the incident rays tend toward Y, the reflected ones will diverge as from X, those which were the 
incident ones in one case, being the reflected ones in the other. 

And lastly, if the incident rays converge toward the centre, they fall in with their respective per¬ 
pendiculars ; on which account they proceed after reflection, as from thence. 

Exr. Illustrate the three preceding propositions by receiving upon a convex mirror, a solar ray 
passing through an open orifice, or a concave or convex lens. 

PROP. LIV. When rays fall upon a plane surface, if they diverge, the focus of the 
reflected rays will be at the same distance behind the surface, that the radiant point is 
before it;—if they converge, it will be at the same distance before the surface, that the 
imaginary focus of the incident rays is behind it. 

Case 1. Of diverging rays. Let AB, AC, be two diverging rays incident on the plane surface DE, 
the one perpendicularly, the other obliquely; the perpendicular one AB will be reflected to A proceed¬ 
ing as from some point inline AB produced; the oblique one AC will be reflected into some line 
CF, such that the point G,^nere the line FC produced intersects the line AB produced also, shall be 
at an equal distance from the surface DE with the radiant A. For the perpendicular CH being drawn, 
ACH and HCF will be the angles of incidence and reflection, which being equal, their complements 
ACB and FCE are so too ; but the angle BCG is equal (El. I. 15.) to FCE; therefore in the triangles 
ABC and GBC the angles at C are equal, the side B€ is common, and the angles at B are also equal to 


Chap. III. 


OF REFLECTION 


123 


each other as being right ones; therefore (El. I. 2G.) the lines AB and BG, opposite to the equal angles 
at C, are also equal, and consequently the point G, the focus of the incident rays AB, AC, is at the same 
distance behind the surface, that the point A is before it. 

Case 2. Of converging rays. Supposing FC and AB to be two converging incident rays, CA and 
BA will be the reflected ones, (the angles of incidence in the former case being now the angles of reflec¬ 
tion, and the reverse) having the point A for their focus; but this, from what was demonstrated above, 
is at an equal distance from the reflecting surface with the point G, which in this case is the imaginary 
focus of the incident rays FC and AB. What is here demonstrated of the ray AC, holds equally of any 
other, as Al, AK, &c. 

Schol. The case of parallel rays incident on a plane surface, is included in this proposition ; for in 
that case we are to suppose the radiant to be at an infinite distance from the surface, and then by the 
proposition, the focus of the reflected rays will be so too; that is, the rays will be parallel after reflec¬ 
tion, as they were before. 

PROP. LV. When parallel rays are incident upon a spherical surface, the focus 
(or imaginary radiant) of the reflected rays will be the middle point between the 
centre of convexity and the surface. 

Case 1. Of parallel rays falling upon a convex surface. Let AB, DH, represent two parallel rays plate 7. 
incident on the convex surface BH, the one perpendicularly, the other obliquely; and let C be the cen- Fig. 1. 
tre of convexity ; suppose HE to be the reflected ray of the oblique incident one DH proceeding as 
from F, a point in the line AB produced. Through the point Id draw the line Cl, which will be per¬ 
pendicular to the surface at that point, and the angles DHI and 1HE, being the angles of incidence and 
reflection, will be equal. But HCF is equal (El. i. 29.) to DHI, and CHF (El. 1. 15.) to IHE; where¬ 
fore the triangle CFH is isosceles; and consequently, the sides CF and FH are equal; but supposing 
BH to vanish, FH is equal to FB, and therefore upon this supposition FC and FB are equal; that is, the 
imaginary radiant of the reflected rays is the middle point between the centre of convexity and the 
surface. 

Case 2. Of parallel rays falling upon a concave surface. Let AB, DH, be two parallel rays incident, p] a t e 7. 
the one perpendicularly, the other obliquely, on the concave surface BH, whose centre of concavity is Fig. 2. 
C. Let BF and HF be the reflected rays meeting each other in F; this will be the middle point be¬ 
tween B and C For drawing through C the perpendicular CH, the angles DHC and FHC, being the 
angles of incidence and reflection, will be equal, to the former of which the angle HCF is equal, as alter¬ 
nate ; and therefore the triangle CFH is isosceles. Wherefore CF and FH are equal; but if we suppose 
BH to vanish, FB and FH are also equal, and therefore CF is equal to FB ; that is, the focal distance of 
the reflected rays is the middle point between the centre and the surface.* 

Schol. The converse of these two cases may be demonstrated in a similar manner, by making the 
incident rays the reflected ones. 

PROP. LVI. When rays fall upon any spherical surface, if they diverge, the dis¬ 
tance of the focus of the reflected rays front the surface is to the distance of the radiant 
point from the same (or, if they converge, to that of the imaginary focus of the incident 
rays) as the distance of the focus of the reflected rays from the centre is to the distance 
of the radiant point (or imaginary focus of the incident rays) from the same. 

* It is here observable, that the farther the line DH is taken from AB, the nearer the point F falls to the surface. For 
the farther the point H recedes from B, the larger the triangle CFH will become ; and consequently, since it is always an 
isosceles one, and the base CH, being the radius, is every where of the same length, the equal legs CF and FH will length¬ 
en ; but CF cannot grow longer unless the point F approach toward the surface. And the farther H is removed from B, 
the faster F approaches to it. This is the reason, that whenever parallel rays are considered as reflected from a spherical 
surface, the distance of the oblique ray from the perpendicular ray is taken so small with respect to the focal distance of 
that surface, that without any physical error, it may be supposed to vanish. Hence it follows, that if a number of parallel 
rays, as AB, CD, EG, kc. fall upon a convex surface, and if BA, DK, the reflected rays^jthe incident ones, AB, CD, pro¬ 
ceed as from the point F, those of the incident ones CD, EG, namely, DK, GL, will pn^^R as from N, those of the incident 
ones, EG, HI, as from O, k c. because the farther the incident ones CD, EG, &c. are Wn AB, the nearer to the surface are 
the points F,/,/, in the line BF, from which they proceed after reflection ; so that properly the imaginary radiants of the p] a * e ri 
reflected rays B A, DK, GL, fcc. are not in the line AB produced, but in a curve line passing through the points F, N, O, Sic. y 

The same is applicable to the case of parallel rays reflected from a concave surfape, as expressed by the dotted lines 
on the other half of the figure, where PQ, RS, TV, are the incident rays; QF, S f. \ j, the reflected ones intersecting each 
other in the points X, Y, and F ; so that the foci of those rays are not in the line FB, but in a curve passing through those 
points. 


124 

Plate 7. 
Fig. 4. 


Plate 7. 
Fig. 4. 


Plate 7. 
Fig. 5. 


Plate 7. 
Fig. 5. 


Plate 7. 
Fig. 6. • 


OF OPTICS. Book VI. 

Case 1. Of diverging rays falling upon a convex surface. Let RB, RD, represent two diverging 
rays flowing from the point R as from a radiant, and falling the one perpendicularly, the other obliquely, 
on the convex surface BD, whose centre is C. Let DE be the reflected ray of the incident one RD ; 
produce ED to F, and through R draw the line RH parallel to FE, till it meets CD produced in H. Then 
will the angie RHD be equal (El. I. 29.) to EDH the angle of reflection, and therefore equal also to 
RDH, which is the angle of incidence ; wherefore the triangle DRH is isosceles, and consequently DR 
is equal to RH. Now the lines FD and RH being parallel, (El. VI. 2.) FD is to RH, or its equal RD, 
as CF to CR; but BD vanishing, FD and RD differ not from FB and RB; wherefore FB is to RB also, 
as CF to CR ; that is, the distance of the imaginary radiant from the surface is to the distance of the 
real radiant point from the same, as the distance of the imaginary radiant from the centre is to the dis¬ 
tance of the real radiant from thence. 

Case 2. Of converging rays falling upon a concave surface. Let KD and CB be the converging in¬ 
cident rays, having their imaginary focus in the point R, which was the radiant in the foregoing case. 
Then as RD was in that case reflected into DE, KD will in this be reflected into DF; for since the 
angles of incidence in both cases are (El. I. 15.) equal, the angles of reflection will be so too ; so that F 
will be the focus of the reflected rays ; but it was there demonstrated, that FB is to RB as CF to CR, 
that is, the distance of the focus from the surface is to the distance (in this case) of the imaginary focus of 
the incident rays, as the distance of the focus from the centre is to the distance of the imaginary focus 
of the incident rays from the same. 

Case 3. Of converging rays falling upon a convex surface, and tending to a point between the im¬ 
aginary radiant of parallel rays and the centre. Let BD represent a convex surface, whose centre is C, 
and focus of parallel rays is P ; and let AB, KD, be two converging rays incident upon it, and having 
their imaginary focus at R, a point between P and C. Now because KD tends to a point between the 
focus of parallel rays and the centre, the reflected ray DE will diverge from some point on the other 
side of the centre, suppose F ; as was shown Prop. LHI. Through D draw the perpendicular CD, and 
produce it to H ; then will KDH and PIDE be the angles of incidence and reflection, which being equal, 
their vertical ones RDC and CDF will be so too, and therefore the vertex of the triangle RDF is besected 
by the line DC ; whence (El. VI. 3.) FD and DR, or, BD vanishing, FB and BR are to each other as FC 
to CR; that is, the distance of the imaginary radiant of the reflected rays is to that of the imaginary 
focus of the incident ones, as the distance of the former from the centre is to the distance of the latter 
from the same. 

Case 4. Of diverging rays falling upon a concave surface, and proceeding from a point between the 
focus of parallel rays and the centre. Let RB, RD, be the diverging rays incident upon the concave 
surface BD, having their radiant in the point R, the imaginary focus of the incident rays in the foregoing 
case. Then as KD was in that case reflected into DE, RD will now be reflected into DF. But it was 
there demonstrated that FB and RB are to each other as CF to CR ; that is, the distance of the focus is 
to that of the radiant; as the distance of the former from the centre is to the distance of the latter from 
the same.* 

Schol. 1. If the reflected* ray be made the incident one, those cases which are respectively the con¬ 
verse of the foregoing may be demonstrated in the same manner. 

Schol. 2. Let it be required to find the distance of the imaginary radiant of diverging rays incident 
upon a convex surface, whose radius of convexity is 5 parts, and the distance of the real radiant from 
the surface is 20. 

Call the focal distance sought x ; then will the distance of the focus from the centre be 5 — rr, and that 
of the radiant from the same 25; therefore (by Prop. LV1.) we have the following proportion, a;: 20:: 5— 
,r: 25, and x = . 

If in any case it should happen, that the value of x should be a negative quantity, the focal point 
must then be taken on the contrary side of the surface. 

* Iti the case of diverging rays falling upon a convex surface, the farther the point D is taken from B, the nearer the 
point F, the imaginary radiant of (he reflected rays, approaches to B, while the radiant R remains the same. For it is evident 
from fhe curvature of a circle, that the point D may be taken so far from B, that the reflected ray DE shall proceed as from 
F, G, H, or even from B, or fronfttev point between B and R, and the farther it is taken from B, the faster the point, from 
which it proceeds, approaches tov^BlR. The like is applicable to any of the other cases of diverging or converging rays 
incident upon a spherical surface, uhis is the reason that, when rays are considered as reflected from a spherical surface, 
the distance of the oblique rays from the perpendicular one is taken so small, that it may be supposed to vanish. From 
hence it follows, that if a number of diverging rays are incident upon the convex surface BD at the several points B, D, D, 
kc. they shall not proceed after reflection as from any point in the line RB produced, blit as from a curve line passing 
through the several points FSic. The same is applicable in all the other cases. 


Chap. III. 


OF REFLECTION. 


125 


SECTION II. 

Of Images 'produced by Refection. 

PROP. LVII. When any point of an object is seen by reflected light, it appears in 
the direction of that line which the ray describes after its last reflection. 

Since reflection gives the same direction to the rays as if they had originally come from the place 
from which the reflected rays diverge, an object seen by reflection must appear in that place. The 
visible image must therefore consist of imaginary radiants diverging from thence. 

PROP. LVIII. In all mirrors, plane or spherical, the place of the imaginary radiant, 
when it is determined, is the intersection of the perpendicular from the radiant to the 
mirror, and any reflected ray. 

Let D be a radiant in any object DE, and DF a ray from this radiant reflected in the line FC ; draw Plate 7. 
DI the perpendicular from the radiant to the mirror, and produce CF, DI, till they meet in L; this point F 'g 7 * 
will be the imaginary radiant. Because the ray DI falls perpendicularly upon the mirror, it will be re¬ 
flected hack in the same line ID, and therefore will appear to come from some point in DI produced. 

And since (from Prop. LVII.) all rays which diverge from the same real radiant before reflection, must 
diverge from the same imaginary radiant after reflection, any other ray from D, as FC, must appear to 
diverge from the same imaginary radiant with the ray DI, that is, from some point in DI; but the ray 
FC (by Prop. LVII.) appears after reflection to proceed in the line FC ; it must therefore appear to come 
from some point in FC, and also from some point in DI, that is, from the point L, in which DI intersects 
FC. The imaginary radiant of the rays DI, DF, after reflection is therefore L, the intersection of the 
perpendicular and the reflected ray. 

Def. XXI. The passage of refection is the incident ray added to the reflected ray ; 
as DF + FC. 

PROP. LIX. In plane mirrors, the distance of the last image from the mirror is 
equal to the distance of the object from it, and the distance of any point in the last im¬ 
age from the eye is equal to the passage of reflection. 

The distance of the imaginary radiants L, M, behind the mirror, are (by Prop. LIV.) respectively Plate 7, 
equal ;o the distances of the corresponding real radiants D, E, before the mirror; therefore the dis- Fig. 7. ’ 
tance of the last image L, M, made up of imaginary radiants between L and M, corresponding to real 
radiants in the object DE, is equal to the distance of that object. The distance of L, the highest point 
of the image, from C, any given place of the eye, is CFL, equal to DFC the passage of reflection, be¬ 
cause (by Prop. LIV.) LF is equal to DF. The same may be shown of M, or any other point in the 
image. / r 

PROP. LX. In plane mirrors, the image is equal and similar to the object. 

If D be the highest point of the object, the highest point of the image is (by Prop. LVII.) in the per¬ 
pendicular DIL ; and ifE be the lowest point of the object, the lowest point of the image is in the per¬ 
pendicular EZM. But DIL and EZM are parallel (El. XI. 6.) because they are both perpendicular to the 
plane surface AB. Consequently, the distance between these lines, that is, the heights of the object and 
image, DE, LM, are equal. In like manner it may be shown, that any diameter of the object is equal 
to its corresponding diameter in the image : whence the object and image are in all respects equal, and 
cortsequeniiy similar surfaces. 


126 


OF OPTICS. 


Book VI. 


. . 

CHAPTER. IV. 

Of Vision . 

SECTION I. 

OF THE LAWS OF VISION. 


Plate 7. 
Fig. 10. 


* 

.Fig. 11. 


PROP. LXI. When the rays which come from the several points in any object enter 
the eye,, they will paint an inverted image upon the retina. 

Let ABA be a section of an eye. AB, BA, is the tunica sclerotica , a white coat w hich encompasses 
the globe of the eye, except the fore part between A and A. The fore part AA is covered by a trans¬ 
parent coat, a little more protuberant than any other part of the eye called the tunica cornea. In the 
cavity of the eye is placed a convex lens C c, consisting of a hard transparent substance, called the 
crystalline humour. This humour is kept in its place and lixed to the coats by certain ligaments all around 
it at e e, called ligamenta ciliaria. Under the tunica cornea , and at some little distance from it, is the iris , 
o, o, which has in it a small oriiice, called the pupil of the eye. This iris is tinged with a variety of 
colours, from which the eye is said to be blue, hazel, black, &c. It consists of muscular fibres, which 
can contract or dilate the pupil. The remaining part between the cornea and the crystalline humour 
is filled wfith a thin transparent fluid, like water, called the aqueous humour. ScN'is a white coat, which 
consists of the fibres of the optic nerve woven together like a net; this coat is called the retina. Between 
the sclerotica and the retina is another coat, which is called the choroides. The cavity of the eye, between 
the crystalline humour and the retina , is filled with a transparent substance, neither so fluid as the aqueous 
humour nor so hard as the crystalline, called the vitreous humour. The optic nerve MBB is inserted at N. 

Mr. Harris has given a table of the dimensions of the human eye, of which the follow ing are the 


principal particulars. 

In. 

Diameter from the cornea to the choroides - .95 

Radius of the cornea.- .335 

Distance of the cornea from the first surface of the crystalline - - .106 

Radius of the first surface of the crystalline - .331 

Radius of the back surface of the crystalline - - .25 

Thickness of the crystalline . .373 


Through the pupil the rays which diverge from the several points of any object ABC pass into the 
cavity. The rays emitted from any point of an object beyond the nearest limit of distinct vision, on 
entering the cornea are rendered converging. The aqueous humour, having about the same density and 
degree of convexity as the anterior surface of the cornea, probably does not much change their conver- 
gency ; but after passing through the pupil they are rendered more and more converging at both surfaces 
of the crystalline humour, and are thus finally thrown upon a single point of the retina. Consequently 
at DEF, or somewhere upon the retina, as upon a piece of white paper in a dark room, an inverted im¬ 
age of the object (by Prop. XXV.) will be painted. 

Schol. The refractive powers of the aqueous and vitreous humours have been found by experiment 
to be about the same as common water; and that of the crystalline is a little greater; that is, the sine 
of incidence is to that of refraction, out of air into the aqueous humour, as 4 to 3, out of the aqueous into 
the crystalline as 13 to 12, and out of the crystalline into the vitreous as 12 to 13. 

Exp. Take off the sclerotica from the back part of the eye of an ox, or other animal, and place the 
eye in the hole of the window shutter of a dark room, with its fore part toward the external objects ; 
a person in the room will, through the transparent coat, see the inverted image painted upon the 
retina. 


Def. XXII. The optic axis, is the axis of the crystalline humour continued to the 
object at which we look. The axis PO of the crystalline humour GPH, continued to 
the point B, is the optic axis directed toward that point. 

D EF. XXIII. That point of the retina , upon which the optic axis continued back 
would fall, is called the middle of the retina . If OP be continued back to E, the point 
F. is the middle of the retina. 


Chap. IV. 


OF VISION. 


127 


PROP. LXII. The images upon the retina are the cause of vision. 

It is found from experience, that when the image upon the retina is bright, the object is clearly seen; 
and when the image is faint, the object appears faint; also, that when the image is distinct, the object 
is seen distinctly; and when the image is confused, the object appears confused. Hence it may be con¬ 
cluded, that these images are the cause of vision. 

Cor. It is manifest that a different conformation of the eye, or some parts of the eye, is necessary 
for distinct vision at different distances. Some think the change is in the length of the eye; others, 
that it is a change in the figure or position of the crystalline humour, and others that it is a change in 
the cornea. Any of these changes would produce the effect. 

PROP. LXIII. The point in any object toward which the optic axis is directed, is 
seen more distinctly than the rest. 

It is known from experience, that when the e} r e is turned directly toward any part of an object, that 
is, when the optic axis is directed toward that point, though the whole object, if it be not very large, 
will be seen, that part on which we look directly will appear most distinct; and the other portions of 
the object, being drawn on parts of the retina , somew hat nearer to the crystalline humour than the mid¬ 
dle point of the retina, will appear a little confused. 

PROP. LXIV. Objects appear erect, although their images on the retina are in¬ 
verted. 

This is known by experience, and is not inconsistent with the explanation above given of vision. 
For it is not the image on the retina , but the object itself, which we see, and w r e judge of its relative 
position, by moving the point of distinct vision along the object, and determine that part to be the high¬ 
est which requires the eye to be the most lifted up in order to see it distinctly. 

PROP. LXV. An object may appear single, although it is seen by both eyes at 
once. 

If both eyes are turned directly to the object C, that is, if the optic axes AC, BC, meet in the ob¬ 
ject. it will appear single. But if, whilst one eye is turned toward C, the corner of the other is pressed 
with the finger so as to alter the position of its optic axis, the object will then appear double. For when 
one eye is turned toward an object, and the other turned a different way, the same object will be seen 
by each eye in a different direction; that is, one eye will see it in one place, and the other in another 
from whence it must appear double. But, if both eyes are directed the same way, that is, to the place 
of the object, though two objects may be said to be seen, yet as both of them are alike, and seen in the 
same place, they will appear but as one. If whilst both eyes are directed toward C, another object D be 
placed at some considerable distance directly beyond it, the object D will appear double; for since the 
eyes see the object D without being turned directly toward it, the place of D is indeterminate ; to the 
right eye it appears on the right hand of C, and to the left eye on the left of C ; that is, being seen in 
two places, it must appear double. If the sight be directed to the farther object D, the nearer object C 
will appear double. For, the object C is seen by the right eye in the direction of a line which passes on 
the left side of D, and by the left eye in the direction of a line which passes on the right of 1). In both 
cases, one of the objects appears double, when the eyes are not directly fixed upon it, that is, when the 
optic axes do not meet in it; and the other object appears single, when the eyes are both directly fixed 
upon it, that is, when the optic axes meet in the object. 

Exr. I. View a nearer and a more distant object at the same time, according to the proposition. 

2 . Let a pasteboard, having a hole in it, be fixed between the eyes of a spectator and two candles so 
placed, that when the right eye is shut, the left eye may see only the one candle, and w'hen the left eye 
is shut, the right eye can see only the other ; although both candles are visible, if both eyes be fixed 
steadfastly upon the hole, they w r ill appear as one candle placed at the hole.. 

PROP. LXVI. There is one part of the retina where no perception of the object is 
conveyed to the mind by the image formed upon it. 

This is found by experiment. Place two small circles of white paper upon a dark coloured wall at 
the height of the eye, and at the distance of near two feet from each other. If the spectator at a proper 
distance, shuts his right eye and looks with the left directly at the paper on his right hand, he will not 
*ce the left hand paper, although the objects round it are visible. Hence it is to be inferred that the 
rays from the left hand paper fall upon a part of the retina which is insensible. 


Plate 7. 
Fig. 12 


128 


OF OPTICS. 


Book. VI. 


Plate 7. 
Fig. 11. 


Plate 7. 
Fig. 11. 


Schol. It is supposed that this part of the retina is that where the optic nerve is inserted ; and be¬ 
cause the coat called the choroides touches the retina in all other parts, but is discontinued here, it has 
been conjectured that the seat of vision is not the optic nerve, but the choroides ; but this point remains 
undetermined. 

Cor. Hence an object cannot become invisible to both e} r es at once ; because the image cannot fall 
upon the optic nerve of each eye at the same time. An object seen with both eyes, is said to appear 
about or brighter than with one eye alone. Harris’s Optics, p. 116. 

PROP. LXVII. If the crystalline humour has either too much or too little convex¬ 
ity, the sight will be defective. 

In persons who are shortsighted, the humours of the eye are too convex, and bring the rays to a 
focus before they reach the retina, unless the object be brought near to it; in which case (by Prop. 
XXVII.) the image is cast farther back. In others, the humours of the eye have so little convexity, 
that the focal point lies behind the retina; whence, unless the object is removed to a great distance 
from the eye, the vision will be indistinct. 

Def. XXIV. The optic angle in viewing any object is the angle at the eye sub¬ 
tended by the diameter of the object. The angle AOC, subtended by AC, the diame- 
of the object, is the optic angle. 

PROP. LXVIII. The apparent diameter of any object is proportional to the diam¬ 
eter of the image of that object on the retina. 

To an eye placed at O, the apparent magnitude of the object ABC is that visible extension which 
lies between A and C. If two lines AO, CO, are drawn from these points crossing one another at the 
eye, they will be the axes of the pencils which come from A and C, and will contain between them the 
diameter AC ; and the points A, C, will be represented on the retina (by Prop. LX1.) at F and D; con¬ 
sequently, DF will be the diameter of the image ; and this diameter is contained between the two lines 
AO, CO, produced to the retina. Now it is manifest, that the visible extension contained between AO 
and CO, that is, the apparent diameter of the object, is as the angle AOC; and that the diameter of the 
image contained between DO and FO is as the angle DOF. But the angles DOF, AOC, (El. I. 15.) are 
equal. Therefore the apparent diameter of the object, and the diameter of the image, are each of 
them proportional to the same angle, and consequently proportional to each other. 

Schol. 1. When we speak of an optic angle, it is not meant that we see the point in which the op¬ 
tic axes meet; but, since by experience we learn to judge of such distances as are not very great, by 
the sensations accompanying the different inclinations of the eye, which are analogous to the optic an¬ 
gle, we express these different inclinations of the eye, by that angle. In like manner, although the eye 
does not see a pencil of rays, whilst the breadth of the pupil bears a sensible proportion to the distance 
of the focus from which the rays diverge to the eye, we have sensations from which experience enables 
us to judge of the place of that focus. So, the magnitude of an image upon the retina being always 
proportional to that of the visual angle of the object, though that angle is not actually measured by the 
eye, the difference of sensations accompanying different magnitudes of the image enable us to distinguish 
different visual angles. Thus it appears, that the use of lines and angles in optics, has its foundation in 
nature. 

Schol. 2. The angle subtended by the least visible object, called by the writers on optics the mini¬ 
mum visibile , cannot be accurately ascertained, as it depends upon the colour of the object, and the 
ground upon which it is seen; it depends also upon the eye. Mr. Harris thinks the least angle for any 
object to be about 40"; and at a medium not less than two minutes. 

To the generality of eyes the nearest distance of distinct vision is about 7 or 8 inches. Taking 8 
inches for that distance, and 2 minutes for the least visible angle, a globular object of less than part 
of an inch cannot be seen. Harris’s Optics, p. 120—124. 

PROP. LXIX. When the diameter of an object is given, its apparent diameter is 
inversely as its distance from the eye. 

The apparent diameter of an object (Prop. LXVIII.) is as the diameter of its image upon the retina.; 
and (Prop. XXXI.) the diameter of the image, when the object is given, is inversely as the distance of 
the object. Therefore the apparent diameter of the object is also inversely as the distance of the ob¬ 
ject. The same may be proved of any apparent length whatsoever. 

Cor. 1 . Hence the apparent diameter of an object may be magnified in any proportion; for the less 


Chap. IV. 


OF VISION. 


129 


the distance of the eye from the object, the greater will be its apparent diameter. But without the 
help of glasses, an object brought nearer the eye than about live inches, though it appears larger, will 
at the same time be seen confusedly. 

Cor. 2. Hence parallel lines, as the sides of a long room, or two rows of trees, as ABC, DEF, seen Plate 7. 
obliquely, appear to converge more and more, as they are farther extended from the eye ; for the * ‘S- *5. 
apparent magnitude of their perpendicular intervals, as AD, BE, CF, &c. are perpetually diminished. 

Cor. 3. An horizontal plane AI seems to ascend. For the visual rays cut a perpendicular DA to the 
horizon, in points that are higher and higher, or nearer to the horizontal line OG, according as they 
proceed from points in AI that are more remote from A. 

Cor. 4. It is for a like reason that a ceiling DH appears to descend, and that faster than the floor 
ascends, as the distance of the eye of the spectator from the ceiling is greater than the distance of the 
eye from the floor. 

PROP. LXX. The apparent diameter of an object, whose distance is given, is di¬ 
rectly as its real diameter. 

The apparent diameter of an object (by Prop. LXVIII.) is as the diameter of its image ; and the di¬ 
ameter of the image (by Prop. XXXIII.) when the distance of the object is given, is as the diameter 
of the object. Therefore the apparent diameter of an object, whose distance is given, is as its real di¬ 
ameter. 

PROP. LXXI. The apparent diameters of different objects at different distances 
from the eye will be equal, if their real diameters are as their distances. 

For (by Prop. XXXIV .) the diameters of their respective images upon the retina will be equal; and 
their apparent diameters (by Prop. LXVIII.) are as the diameters of their images. 

PROP. LXXII. The apparent length of an object, seen obliquely, is as the appar¬ 
ent length of a subtense of the optic angle perpendicular to the optic axis. 

If DF be an object seen obliquely, and DG an object seen directly, that is, if DF be oblique, and DG ?. 
perpendicular to the optic axis OR, then supposing them to subtend the same angle DOF, their images * 
upon the retina (Def. XXIV.) will be equal, whence (by Prop. LXVIII.) their apparent diameters will 
be equal. Consequently, the greater the subtense GD is, the greater will be the apparent length of the 
object DF; and the reverse. 

Cor. Hence an object appears shortened by being seen obliquely. 

PROP. LXXIII. When equal objects in the same right line are seen obliquely, 
their apparent lengths are inversely as the squares of their distances from the eye. 

Let the eye be at O ; and in the line BC at different distances from the eye take equal spaces DF, Plate 7. 
df The apparent length of DF (by Prop. LXXII.) is proportional to the apparent length of GD, the 1 '§• 
subtense of the optic angle DOF, perpendicular to the optic axis OR. In like manner, the apparent 
length of df is proportional to that of g d, the subtense of d O f. Now GD, gd , are subtenses also of 
the angles GFD, gfd ; and as the side Of is to the side OF, so is the sine of the angle OF f that is, of 
its supplement OFB, to the sine of the angle O/F, or O/B. Hence, since small angles are to one 
another nearly as their sines, if these are small angles, Of will be to OF, as the angle OFB to the angle 
0/*B ; that is, O /will be to OF, as the subtense GD to the subtense g d, or GD is to g d inversely as 
OF to O f; that is, the subtenses of the optic angles, and consequently, from what has been shown, the 
apparent diameters DF, df are inversely as their distances from the eye. This proportion arises from 
the different degrees of obliquity at which the eye sees the equal spaces DF, df But if their obliqui¬ 
ties with respect to the eye were the same, the apparent length of DF (by Prop. LXIX.) would be to 
that of df inversely as their distances. Since then the apparent lengths of DF, df are inversely as 
their distances on account of their different obliquities, and also inversely as their distances on account 
of their different distances; on both accounts taken together, they are in the ratio compounded of the 
inverse ratio of their distances, and the same, that is, inversely as the squares of their distances. 

PROP. LXXIV. The apparent diameter of an object is not changed by contracting 
or dilating the pupil. 

For when the distance is given, the diameter of the image (by Prop. XXXV I. Schol. 2.) remains the 

17 


130 


Plate 7. 
Fig. 14. 


Plate 7. 
Fig. 14. 


Plate 7. 
Fig. 14. 


OF OPTICS. Book VI. 

same, whatever be the area of the pupil, and consequently (by Prop. XXXIII.) the apparent diameter 
of the object. 

PROP. LXXV. Aii object appears larger when it is seen confusedly, than when it 
is seen distinctly. 

For the confused image (by Prop. XXIX.) is larger than the distinct image, and consequently (by 
Prop. LXVI1I.) the apparent magnitude of the object is greater when it is seen confusedly than when it 
is seen distinctly. 

Cor. Hence objects appear magnified when seen through a mist; the drops of which refract the 
rays so differently, that they cannot be collected into one focus. 

PROP. LXXVI. A spectator in motion sees an object at rest, moving the contrary 
way. 

If while an object at P is* at rest, the eye be carried parallel to PQ in the direction from Q toward 
P, its image on the retina will move from p to q ; the same effect will be produced, if the object move 
from P toward Q; and if the velocity of the object and the eye, in each case, be the same, the appar¬ 
ent velocity will be the same also. 

Cou. 1 . An object moving along a line PK will appear at rest to a spectator moving along the line 
QG, parallel to PK; if the motion of the object be quicker or slower than that of the spectator, it will 
have an apparent motion direct or retrograde ; if the two motions are in contrary directions, the appar¬ 
ent motion of the object w ill vary r with the real motion of the spectator. 

Cor. 2 . If the earth be supposed to move round its axis from west to east, while the heavenly bodies 
are at rest, they appear to us to move the contrary way, there being nothing in this case to indicate to 
us our own motions. And therefore no argument drawn from the apparent diurnal motions of the stars 
and planets can be of any support to either the Ptolemaic or Pythagorean systems. 

Schol. A person riding, or walking slowly, though he perceives the change of situation of adjacent 
bodies, yet being sensible of his own motion, and having time to reflect in the intervals of these appar¬ 
ent changes, those bodies appear to keep their places. But if he runs or rides very swiftly, he cannot 
help fancying, that the bodies, which he is looking at, are moving tow'ard him. The deception is still 
stronger when he sits at his ease in a swift sailing vessel. 

PROP. LXXVII. The same degree of velocity appears greatest, when the motion is 
in a line perpendicular to the optic axis; and when the motion is in other directions, 
the apparent velocity will be as the cosine of the angle of inclination to the said per¬ 
pendicular. 

If two bodies set out at the same time from P, the one moving along the line PQ, perpendicular to 
the optic axis Q*y, and the other along the line PS, oblique to it, and if their velocities be such, as to 
pass over the lines PQ, PS, in the same time, it is manifest, that their apparent velocities will be the 
same ; for the images of each w ill pass over the same space p < 7 , on the retina , in the same time. The 
real velocities being, in this case, as PQ to PS, it is manifest, that when the velocities are equal, the 
apparent velocity of the body which moves in PQ is to that of the body moving in PS. as PS to PQ ; 
that is, as radius to the cosine of the angle QPS ; but PS is always greater than PQ ; whence the propo¬ 
sition is manifest. 

PROP. LXXVIII. If objects at different distances from the eye move in parallel 
lines, nearly at right angles to the optic axis, and if their velocities are proportional to 
these distances, their apparent velocities will be equal; and if their real velocities are 
equal, their apparent velocities will be reciprocally as their distances from the eye. 

Let an object move from Q to P, in the same time that another moves from G to H, their real ve¬ 
locities are as QP to GH, that is, (El. VI. 2.) as QO to GO, the distances from the eye ; and their ap¬ 
parent velocities will be equal; for the space qp upon the retina will be passed over in the same time 
by the image of each. If the velocities of the objects G, Q, be equal, the object G will arrive at K, and its 
image describe the space q k upon the retina , in the same time that the image of the object Q describes 
the space qp ; w r hence the apparent velocities of these two objects are as qk and q p, or as GK (or QP) 
and GH; that is, (El. Vl. 2.) the apparent velocity of the object G is to that of Q, as QO 
to OG. 

Schol. It is here supposed, that the spectator makes no allowance for the different distances. 


Chap. IV. 


OF VISION. 


131 


PROP. LXXIX. The apparent velocities of bodies moving in parallel lines at dif¬ 
ferent distances from the eye, are directly as the real velocities, and reciprocally as 
the distances. 


Let two bodies move from Q, G, in parallel lines QP, GH ; let the velocity of the object Q be called 
V, and that of G, v ; and lei their apparent velocities be called A, a. If the two objects be conceived 
to move in the same line GK, whatever be their velocities, V: v : : A and supposing the velocity of 
the object Q, to be the same in QP as before in GK, A in QP : A in GK;: GO : QO, by Prop. LXXVIII. 

V V 

and A in GK : a :: V : v ; whence, compounding these ratios, A : a : : GO X V : QO x v, that is,: : — — . 

GO 

Schol, 1 . We judge of the distance of any object by the visible length of the plane which lies between 
the eye and the object. When this method fails us, we compare the known magnitude of the object, 
with its present apparent magnitude ; or we compare the degrees of distinctness with which we see the 
several parts of an object; or we observe whether the change of the apparent place of an object when 
viewed from different stations, or its parallax , be great or small, this change being always in proportion 
to the distance of the object. On this principle, we may judge of the distance of a near object by 
observing the change which is made in its apparent situation, upon viewing it successively, with each eye 
singly. Or since it is the difference of the apparent place of an object, as viewed by each eye separately, 
which makes an object to be seen double unless we turn both eyes directly toward it, and since in doing 
this, where the distance is very small, we turn the eyes very much toward each other, and less at a 
greater distance ; the different sensations accompanying the different degrees in which the eyes are turn¬ 
ed toward each other, afford, by habit, a rule for judging of the distance of objects. 

Schol. 2 . In objects placed at such distances as we are used to, and can readily allow for, we know 
by experience how much an increase of distance will diminish their apparent magnitude , and therefore 
instantly conceive their real magnitude, and neglecting the apparent, suppose them of the size they 
would appear if they were less remote; but this can only be done, where we are well acquainted with 
the real magnitude of the object; in all other cases, we judge of magnitudes by the angle which the 
object subtends at the known, or supposed distance; that is, we infer the real magnitude from the appar¬ 
ent magnitude in comparison with the distance of the object. 


SECTION II. 

Of Vision as affected by Refraction. 

PROP. LXXX. When any small object is seen through a refracting medium, it ap¬ 
pears in the direction of that line which the rays describe after their last refraction. 

The ray DE from any small object D, in passing through a glass prism, the end of which is ACB, 
will be refracted toward a perpendicular, and will describe the line EF; and when it goes out of the 
prism, it will be refracted from a perpendicular into the line FG, which is its direction after its last re¬ 
fraction • and the object D will be seen in this direction at L instead of D. For the image in the retina 
will be in the place in which it would have been, if the naked eye had been looking at an object really 
placed at L, the last direction of the rays. 

Cor. Hence an object seen through a glass cut into different surfaces inclined to one another, will 
appear at one view in many different places. The object A seen from the point F through the glass 
CEDB, will appear at H, A, G ; the last direction of the rays AC, AD, AB, after refraction. 

Exp. View any object through a multiplying glass. 

Schol. If a hair be placed across a small hole made in a thin board, and an eye situated, in the dark, 
look through the hole at a number of candles, the hair will appear multiplied; for a shadow of the hair 
is cast upon the eye by each of the candles. 

PROP. LXXXI. An object seen through a denser medium terminated by plane and 
parallel surfaces, appears nearer, brighter, and larger, than with the naked eye. 

Let AB represent the object, CDEF the medium, and GH the pupil of an eye. Let RK, RL, be two 
rays proceeding from the point R, and entering the denser medium at K and L; these rays will here by 
refraction be made to diverge less (by Prop. XVII.) and to proceed afterward, suppose in the lines K a, 
Lb; at a and where they pass out of the denser medium, they will be as much refracted the. con- 


Plale 7. 
Fig. 14. 


Plate 7. 
Fig. 16. 


Fig. 17. 


Plate 7. 
Fig. 18. 


OF OPTICS. 


Book. VI. 


132 

Irary way, proceeding in the lines o c, bd , parallel to their first directions (by Prop. XI.) ; produce these 
lines back till they meet in c, this will be the apparent place of the point R, and it is evident from the 
figure, that it must be nearer the eye than that point; and because the same is true of ail other pencils 
flowing from the object AB, the whole will be seen in the situation /g, nearer to the eye than the line 
AB. As the rays RK, RL, would not have entered the eye, but have passed by it in the directions K r, 
L t , had they not been refracted in passing through the medium, the object appears brighter. The rays 
A A, Bi, will be refracted at A and i into the less converging lines A A, il , and at the other surface into 
k M, l M, parallel to A h aud B i produced (by Prop. XI.), so that the extremities of the object will appear 
in the lines M A, M /, produced, namely, in f and g, and under as large an angle/Mg, as the angle A q B, 
under which an eye at q would have seen it, had there been no medium interposed to refract the rays ; 
and therefore it appears iarger to the eye at GH, being seen through the interposed medium, than other¬ 
wise it would have done. But it is here to be observed, that the nearer the point e appeal's to the eye 
on account of the refraction of the rays RK, RL, the shorter is the image /g, because it is terminated by 
the lines M f and Mg, upon which account the object is made to appear less ; and therefore the apparent 
magnitude of an object is not much argumented by being seen through a medium of this form. 

Farther, it is apparent from the figure, that the effect of a medium of this form depends wholly 
upon its thickness ; for the distance between the lines R r and e c, and consequently the distance be¬ 
tween the points e and R depends on the length of the line 1(«; again, the distance between the lines 
AM and/M, depends on the length of the line A A, but both K a and A A depend on the distance between 
the surfaces CE and DF, and therefore the effect of this medium depends upon its thickness. 

PROP. LXXXII. In viewing objects through a convex or concave lens, the ob¬ 
ject itself is not seen, but its last image, consisting of all the imaginary radiants from 
which the rays appear to diverge after the last refraction. 

Plate 7. If AC be an object nearer the convex lens GIL than its principal focus, the rays which diverge 

F ‘g- 8 . p rom an y point B in this object, will in passing through the lens (by Prop. XVII.) be made to di¬ 
verge less, and the imaginary radiant (by Prop. VI.) will be more distant than the real one. Hence 
the rays BG, BL, after refraction will appear to diverge from the radiant E, farther from the lens 
than the real radiant B. The same happening to the rays from every other point of the object, there 
will be in DEF as many imaginary radiants as there are real radiants in the object ABC, which, taken 
together, will compose the last image. And since all the rays fall upon the eye as if they had di¬ 
verged from this last image, the eye will be affected by the object ABC in the same manner in which 
it would be affected without the lens, by an object in all respects like DEF, that is, the eye per¬ 
ceives the last image. 

PROP. LXXXIII. An object seen through a convex or concave lens will appear 
erect, if the object and its image are on the same side of the lens, but inverted if they 
are on contrary sides. 

Plate 7. It appears from the last Prop, that all the rays which diverge from B before refraction, will appear 

Pig. 8,9. to diverge from E after refraction; and the like may be said of any other points, A and D, or C and F. 

Now A1 is the axis of the beam which comes from A, and therefore, with the rest of the rays of the 
beam after refraction, will appear to diverge from the point D in the same right line with A. The 
same may be shown of FCI. Now these right lines only cross one another at the lens. Consequent¬ 
ly, the highest point both of the object and image is in DI, the upper side of the angle DIF, and the 
lowest points of both in FI, the lower side of the same angle ; that is, the image, which is visible, is 
erect, or in the same position with the object. 

If the object ABC is more remote than the principal focus of the convex lens E, there will be (by 
Prop. XXIV.) a distinct image formed on the other side of the lens. If the rays thus collected are 
not received upon a plane surface, they will proceed straight fonvard ; those which had converged 
diverging from the focus; whence an inverted image will be presented to an eye placed beyond the 
focus.—In the same manner this, and the preceding proposition, may be proved concerning an object 
seen through a concave lens. 

Exp. 1 . Observe the image of a candle whose rays have passed through a convex lens, and are 
received at the focus on a w’hite surface, whilst the object is on the same, or on the contrary side of 
the lens. 

2. An inverted image will be produced without a lens, by solar rays passing through a very small 
hole into a darkened room ; and if the edge of a knife be applied to one side of the hole, and moved 
slowly over it, the parts of the image situated opposite to the covered side will be first concealed : 
from whence it is manifest, that the rays cross one another in passing through the hole. 


Chap. IV. 


OF VISION. 


133 


PROP. LXXXIV. The apparent magnitude of an object seen through a lens 
placed close to the eye, or to the object, is equal to its apparent magnitude when 
seen without the lens. 

If (he eye be placed close to the lens at I, the diameter of the object of refracted vision DF, is Plate 7. 
to the diameter of the object of plane vision AC (by Prop. XXVIII.) as EL to IB, that is, as their Fig. 8. 
respective distances from the vertex of the lens. Therefore (as appears from Prop. LXIX.) their 
apparent diameters, when seen from I, are equal. 

If the lens he placed close to the object, the real radiants touching the lens, the imaginary radiants, 
that is, the last image, will also touch the lens; whence their diameters, or apparent magnitudes will 
he equal. 

PROP. LXXXV. If an object seen through a convex lens is nearer to the lens 
than its principal focus, it will appear brighter than to the naked eye, distinct, and 
in an erect position. 

In this case, the rays which come from any point (by Prop. XVII.) will be brought nearer by re¬ 
fraction, and consequently a greater numljer will enter the eye, than if there had been no refraction ; 
whence it is manifest, that the object will appear brighter. Because the rays of each pencil diverge 
after refraction, but less than before, they come to the eye, as they would if the object were at a 
moderate distance and no lens were used, and therefore will he seen distinctly. And because the refract¬ 
ed rays (by Prop. XVII.) diverge less than the incident rays, that is, (by Prop. VI.) the imaginary 
radiants are more remote than the real ones, the last image, as DEF, which is formed by these 
imaginary radiants, is farther from the lens than the object, and on the same side of the lens ; and 
consequently, since the extreme axes DAI, FCI, only cross one another at the lens, the image will 
be in the same position with the object and appear erect. 

PROP. LXXXVI. If an object seen through a convex lens be farther from the 
the lens than its principal focus, the object will appear brighter than to the naked 
eye, confused, and in an erect position. 

If the eye be placed between the lens and the distinct image, whilst the eye is nearer the lens 
than the place of the image, the rays being made convergent by the lens, will be closer together, and 
therefore a greater number of them will enter the eye, and the object will appear brighter, than if 
it had been seen without a lens; because the rays come to the eye in a converging state, which from 
one and the same point they do not in plane vision, they will give a confused image. And because no 
image is formed till the rays come to the retina , the object will appear erect. 

PROP. LXXXVII. If an object seen through a convex lens be in the principal focus, 
it will appear brighter than to the naked eye, and erect. 

The rays of each particular beam, becoming in this case (by Prop. XVII.) parallel after refraction, are 
brought nearer together than if there had been no lens; consequently, more rays will enter the eye 
from every point, and the object will appear brighter; and no image being formed before therayscome 
to the retina , the object will appear erect. 

PROP. D. A minute object, when seen through a lens of very small principal 
focal length, appears magnified and distinct, if the object be placed in the principal 
focus. 

The angle under which the object appears, will be to that which it subtends, when seen by the 
naked eye ; as the distance at which it is viewed by the naked eye distinctly, is to the principal focal 
length of the lens. 

If the principal focal length of the lens be ^ of an inch, and the distance at which the eye can see 
distinctly be eight inches, it follows that the lens will magnify 240 times in diameter. 

PROP. LXXXVIII. When an object, seen through a convex lens, is nearer than 
the principal focus, it is magnified, unless the lens touches the eye or the object; and 
as the eye departs from the lens, its apparent magnitude will decrease. 

If the object ABC continue in the same place, or do not change its distance BI from the lens, the Flate 7. 
last image DF will (by Prop. XXVII.) remain at the same distance El; therefore the real diameter 


3 34 


OF OPTICS. 


Book VL 


Plate 7. 
Fig. 9. 


of DF (by Prop. XXXIII.) wil be invariable, wherever the eye is placed. If the eye be at the ver¬ 
tex of the lens I, the apparent diameters of the last image DEF, seen through the lens, and that of the 
object ABC, seen with the naked eye, are manifestly the same. Both these apparent diameters de¬ 
crease as the eye recedes from the lens toward O ; but the apparent diameter of the last image de¬ 
creases in the inverse ratio of OE to IE, and that of the object in the inverse ratio of OB to IB. But 
OE has a less ratio to IE than OB to IB; for IE and IB are unequal quantities, of which IE, the dis¬ 
tance of the image, is always (by Prop. XVII. and VI.) the greater, which are equally increased, but 
not proportionally ; therefore the apparent diameter of the image decreases slower than that of the 
object, as the eye recedes from the lens. Consequently, when the eye is at any distance from the 
lens, the last image, or the object of refracted vision, will appear greater than the object seen by the 
naked eye. As the eye departs from the lens, the apparent magnitude of the object, from what has 
been said, must continually decrease, till at an infinite distance it vanishes. 

PROP. LXXXIX. If an object seen through a convex lens is more remote than 
the principal focus, and the eye on the other side of the lens is nearer than the place 
of the image, the object appears magnified, and its apparent magnitude will be in¬ 
versely as the distance of the eye from the image. 

Suppose the eye at the side of the lens GL; it might successively see both the object and the 
image without looking through the lens ; and in this situation (by Prop. XXVIII.) the real diameter 
of the object is to that of the image, as their respective distances from the lens or the eye; conse¬ 
quently (by Prop. XXXIV.) their apparent diameters will be equal. Next, suppose the eye close to 
the image at F, E, or D, the apparent diameter of the image would manifestly be infinite. Also in 
this situation of the eye, the apparent diameter of the object would be infinite ; for, if the eye be at 
F, the rays from the point C are the only rays collected into the eye, which appear diffused over the 
whole surface, and would do so if the lens were ever so large ; and the same would be true of the 
points B or A, if the eye were at £ or D ; that is, the apparent diameter of the object seen through 
the lens is infinite. Since then the object and the image appear equal when the eye is close to the 
lens, and both appear infinite when the eye is close to the image, they must have increased equally as the 
eye was moving from the lens to the image, and their apparent diameters must always have been equal. 
Hence, the object in every station of the eye, when it does not touch the lens, is magnified. And because 
the apparent diameter ofthe object seen through the lens is every where equal to that of the image, and 
that of the image (by Prop. XXXI.) inversely as the distance of the eye from the picture, the apparent 
diameter of the object seen through the lens, is inversely as the distance of the eye from the image. 

PROP. XC. If an object seen through a convex lens be placed in the principal 
focus, its apparent magnitude will not be altered by withdrawing the eye from the 
lens. 

Since in this case the rays from the object come parallel to the eye, both the imaginary radiants 
and the image (by Prop. XVII.) are infinitely distant. Therefore the apparent magnitude of the ob¬ 
ject cannot be diminished by receding from the imaginary radiants, nor increased by approaching to 
the image, but will always remain the same. 

PROP. XCI. If a convex lens be moved whilst the eye and object remain fixed, 
the apparent magnitude of the object will increase, till the lens is at the middle point 
between them, after which it will decrease till the lens reaches the object; provided 
the eye is never farther from the lens than the place of the image. 

When the lens is at either extreme (by Prop. LXXXIV.) the object is not magnified ; but between 
the extremes (by Prop. LXXXV1II. and LXXXIX.) it is magnified; therefore when it is equally 
distant from the two extremes, it is most magnified, and must increase in its apparent magnitude as 
the lens moves from the eye toward the middle station, and decrease, as it moves from that middle 
station toward the object. * 

PROP. XCII. The apparent magnitude of an object, seen through a concave 
lens, decreases as the eye, or the object, departs from the lens. 


* The reasoning here appears not perfectly conclusive. 


Chap. IV. 


OF VISION. 


13 


If the eye touches the vertex of the lens I, the apparent diameters of the object and the last image Plate 7. 
are equal. As the eye recedes from the lens, its distance both from the object ABC and last image Fig. 9, 
DEF increases, and consequently, the apparent magnitude of both decreases. But the distance IE from 
the last image increases faster than the distance IB from the object, as was shown in Prop. LXXXV1II. 
Therefore (by Prop. LXIX,) the apparent diameter of the last image, or the object of refracted vision, 
is diminished as the eye recedes from the lens, more than that of the object seen by the naked eye. 

—Again, as the object departs from the lens, the image departs with it; whence its visible diameter 
decreases. * 

PROP. XCIII. When the eye and object are fixed, if a concave lens be moved 
from the eye, the apparent magnitude of the object will decrease till it reaches the 
middle point between them, and increases as it moves on toward the object. 

When the lens is at each extreme, the apparent magnitude of the object seen through the lens (by 
Prop. LXXXIV.) is the same as when seen with the naked eye. In all other-stations of the lens, the ob¬ 
ject appears diminished ; therefore it must appear most of all diminished when the lens is in the middle 
station, and it must decrease whilst it is approaching to that station, and increase whilst it is departing 
from thence toward the object. 

Exr. View a candle through a convex or concave lens, in the manner described Prop. XLII. varying 
the position of the object, or lens, according to the preceding Propositions, from Prop. LXXXIV. 

PROP. XCIV. Convex lenses assist the sight of those persons whose eyes are not 
sufficiently convex, and concave lenses, that of those whose eyes are too convex. 

For convex lenses enable the former to bring the rays from objects to a focus nearer to the crystalline 
than can be dune by their eyes; and concave lenses enable the latter to bring the rays to a focus at a 
greater distance ; and thus to produce a distinct image upon the retina. 

Schol. Were there no other use of the science of dioptrics, says Mr. Molyneux, than that of spec¬ 
tacles, the advantage that mankind receives thereby is inferior to no other benefit, not absolutely re¬ 
quisite to the support of life. For as the sight is the most noble and extensive of all our senses;—as 
we make the most frequent use of our eyes in all the actions and concerns of life, surely that instrument 
which relieves the eyes when decayed, and supplies their defects, must be estimated as the greatest of 
all advantages. Forlorn must have been the situation of many young, and almost all old people, before 
this admirable invention. The same author concludes, that spectacles were lirst used about the end of 
the 13th century ; and he ascribes the invention to Friar Bacon. 


SECTION III. 

Of Vision as affected by Reflection. 

PROP. XOV. If a plane mirror and the object seen in it are both perpendicular 
to the horizon, the object appears erect. 

The object DE, and the mirror AB, being both perpendicular to the horizon, the lines DI, EZ, in Plate 7. 
which (by Prop. LVII.) the highest and the lowest points of the object appear, being both perpendicular Fig. 7. 
to the surface AB, are parallel to each other, and do not meet. Therefore the line DI, which is high¬ 
est at the object, is also highest at the image, and EZ will be the lowest at both ; therefore the image 
is not inverted with respect to the object; and each point of the image LM (by Prop. LIV.) is equally 
distant from the surface of the mirror with its corresponding point in the object DE ; therefore LM, DE, 
are parallel (El. I. 30.), and since the object is erect, tire image will be so too. 

PROP. XCVI. When the object is parallel to a plane mirror, the length or breadth 
of that part of the mirror, upon which the image appears, is to the length or breadth 
of the object, as any rellected ray is to the passage of reflection. 

* The reasoning here appears not perfectly conclusive. 



136 


OF OPTICS. 


Cook. VI. 


Plate T. 
Pig. 7. 


Plate 7. 
Fig 7. 


It' the object DE is parallel to the mirror AB, and the image LM is seen by the eye at C, then FN. 
the length of that part of the mirror which is taken up by the image, subtends the angle LCM, under 
which the image appears. For since all the visible length of the image is manifestly included within 
the angle LCM, there cannot be more of the mirror taken up by that visible length, than is included 
within the same angle. Now the length of the image LM is equal (by Prop. LX.)to the length of thp 
object DE. And (El. VI. 2.) FN is to LM, as FC to CL, or (by Prop. LIX.) CFD; that is, the length 
of that part of the mirror which is taken up by the image is to the length of the image, or (by Prop. 
LX.) the length of the object, as any reflected ray is to the passage of reflection of that ray. In the 
same manner it may be shown, that the breadth of that part of the mirror taken up by the image, is to 
the breadth of the object in the same ratio. 

Cor.. Hence, in plane mirrors the object, and the part of the surface on which it appears, are 
similar. 

PROP. XCVTI. If, at a certain distance from the mirror, the whole object cannot 
be seen by reflection, the whole will become visible either by bringing the eye nearer to 
the mirror, or removing the object farther from it. 

For (by Prop. XCVI.) the less the ratio of the reflected ray is to the passage of reflection, so much 
the less will be the ratio of the length of that part of the mirror on which the whole object will appear, 
to the length of the object. If therefore the reflected ray CF decreases by bringing the eye nearer to 
the mirror, since it is diminished faster than the greater quantity DFC, the passage of reflection, when 
equal parts are taken from each, the ratio of the reflected ray CF to the passage of reflection DFC (El. 
V. 8.) diminishing, the ratio of the length of that part of the mirror, on which the whole object will be 
seen, to the object, is diminished ; that is, (the object being given) the length of the part of the mirror 
on which the whole object is seen, will be diminished; consequently, the whole object, which at a cer¬ 
tain distance of the eye from the mirror was not visible, on a nearer approach of the eye may become 
visible. 

If the object DE be removed farther from the glass, the ratio of the reflected ray FC to the 
passage of reflection DFC will also be diminished, because DFC will be increased whilst FC remains 
the same ; and consequently the length of the part of the mirror, on which the whole image is seen, 
is diminished, and a less surface of glass is required in order to see the whole image. 

PROP. XCVIII. If a spectator sees himself entirely in a plane mirror placed 
parallel to him, the mirror must be half as long as himself. 

When a spectator is looking at himself, the incident ray is his distance from the mirror, and the 
reflected ray is equal to it, and is the distance of the mirror from him. The passage of reflection is 
therefore equal to twice his distance from the mirror; and consequently the reflected ray is to the 
passage of reflection as 1 to 2 ; whence (by Prop. XCVI.) the length of the glass, in which he can 
see himself entirely, must be to his own length as 1 to 2, or the length of the glass must be half his 
own length. 

If the mirror be at all shorter than this, the spectator will not be able to see himself, whether he 
Is nearer to the glass or farther from it. If he approaches toward the glass, the object, being himself, 
approaches as fast as the eye, so that, though (by Prop. XCVJI.) he might see more of himself by the 
approach of the eye, he will see just as much less of himself on account of the approach of the object. 
In the same manner it may be shown, that if he recedes from the mirror, he will not be able to see 
himself entirely. 

i PROP. XCIX. Objects perpendicular to the horizon, seen in a plane mirror par¬ 
allel to the horizon, appear inverted. 

By Prop. LIV. each imaginary radiant is at the same distance behind the mirror, that the real 
radiant is before it; hence, if the mirror be below the object and the eye, the object will have its 
lowest part nearest the surface of the mirror, and its highest part farthest from it, and therefore will 
in this situation appear inverted; if the mirror be above the object and the eye, the object will have 
its highest part nearest the surface, and its lowest part farthest from it, and therefore will, in this 
situation also, appear inverted. 

PROP. C. If a plane mirror be inclined to the horizon at an angle of forty-five 
degrees, an object parallel to the horizon will appear erect in the mirror, and an ob¬ 
ject perpendicular to the horizon will appear parallel to it. 


Chaf. IV. 


OF VISION. 


13 


Let the object CD parallel to the horizon, be seen in AB, a mirror so placed as to incline to the rinte 7. 
horizon in an angle of forty-five degrees. At whatever distance any radiant C is from the mirror, at Fig. 19. 
the same distance (by Prop.LlV.) is the corresponding radiant c in the image, or CE will be equal to c E. 

In like manner, 1)13 will be equal to d 13. Thus every radiant in the image is at the same distance 
behind the mirror, as the object is before it; whence the image c cl makes half a right angle c BA with the 
mirror on one side, whilst the object makes with it half a right angle CBA on the other; whence c!3C 
is a right angle ; that is, the image is perpendicular to the object or horizon, and appears erect in the 
mirror. 

By making c d the object, and CD the image, it may be shown in like manner that when the object 
is erect, it will appear in the mirror parallel to the horizon. 

PROP. Cl. If an object be placed between two plane mirrors inclined to one an¬ 
other at any angle, several images may be seen. 

Let the object F be placed between the two plane mirrors CB, CA, making with each other the Plate 7. 
angle BCA. From the object F, draw FD perpendicular to the mirror CA, meeting CA in K, and Fi S- 2 
make KD equal to FK. The image of F will (by Prop. L1V.) appear at D. In like manner, if FG 
be drawn perpendicular to CB, the object will be seen in G, as far behind the mirror as F is before 
it. Thus two images of the same object, but of different sides or surfaces of it, will be seen. 

Again, since some of the reflected rays, which diverge from the image D in all directions fall upon 
the opposite mirror CB, the image D may be considered as an object placed before tue mirror CB ; 
and consequently, when the rays which diverge from D are reflected from CB, if DHE be drawn 
perpendicular to CB, and if EH is taken equal to HD, these reflected rays will (by Prop. LVIII.), 
represent the image of D at E, as far behind the mirror as D is before it. In like manner, some of 
the rays from this second image E will fall upon the mirror CA, and the image of E, or a third image of 
the object, will appear. And thus, as long as the image represented in one mirror is before the other, 
so long a new image of the last image will be produced. And all these images, beginning from CA, 
and being successive representations of D, will be images of the side of the object F toward CA. 

Besides these, there will be another set of images beginning from CB, which will be formed in the 
same manner, and represent the side of the object toward CB, the first of which will be G, and the 
second L. 

PROP. CII. The images which appear in two plane mirrors inclined to one an¬ 
other, are in the circumference of a circle, the radius of which is the distance of the 
object from the vertex of the angle contained between the mirrors. 

Since FK is equal to KD, KC common, and the angles at IC right angles, CF, the distance of the Plate 7 
object from the vertex of the angle made by the inclination of the mirrors to one another, is (El. I. 4.) F *g- 20 - 
equal to CD. In like manner it may be proved that CE is equal to CD. Therefore CF and CE 
are equal to one another. Thus all the straight lines drawn from C to G, L, or any other image in 
the mirrors, may be proved to be equal to CF. Consequently, if C be made the centre of a circle, 
and CF its radius, the circumference will pass through the points D, E, G, L, and every other image 
which appears in the mirrors, that is, all the images are in the circumference of a circle, whose 
radius is the distance of the object from the vertex of the angle made by the inclination of the mir¬ 
rors to one another. 

PROP. CIII. If two plane mirrors are parallel to one another, and an object is 
placed between them, innumerable images of that object may be seen in each, stand¬ 
ing in a right line. 

If the two mirrors CA, CB, were separated at C, so as to be less inclined, or more nearly parallel to p| a(e - 
one another, the angle of inclination being diminished, it is manifest from Prop. CII. that, the number Fig. 20. 
of images will be increased. At the same time the circumference of the circle in which the ima¬ 
ges are placed will be enlarged, because the vertex C is farther removed from F, or FC is increased. 
Consequently, if the mirrors CA, CB, are so far separated, that the vertex is infinitely distant, the 
images become innumerable, and they are placed in a straight line. 

PROP: CIV. In spherical mirrors, concave or convex, when the place of the im¬ 
age is determinate, the object and the image are in the same situation, if they are 
both on the same side of the centre, and in contrary situations if they are on opposite 
sides. 

1 O , 

11 / ' 


138 


Plate 7. 
Fig. 23. 


Plate 7. 
Pig. 22 . 

Fig. 21. 


Plate 7 
Fig. 21 


OF OPTICS. 


Book VI. 


Let AFB be an object placed nearer the concave mirror SGV than its principal focus, and let C 
be the centre of concavity. The rays from A, F, B, being reflected, will (by Prop. Lll.) diverge, and 
the distances of the corresponding imaginary radiants 1, E, M, may be determined by Prop. L\ 1. The 
real and imaginary radiants are, in this case, on the same side of C the centre ot concavity. Xow, 
the imaginary radiant which corresponds to the real one A, is (by Prop. L\ til.) in CAl the perpen¬ 
dicular drawn from A to the surtace : and the same with respect to B. and all the other radiants. And 

these perpendiculars, and all the rest, being drawn from the centre, do not cross each other but at 
the centre; consequently they are in the same position with respect to each other at the object and 
the image, and that which is the highest at the object will be the highestatthe image, and the reverse. 
Since therefore (by Prop. LVIII.) every point of the object appears in its perpendicular at the image, the 
highest point in the object will appear the highest in the image, and the reverse ; that is, the object and 
image will be in the same situation. In like manner it may be shown, that if the object AFB is placed 
before a convex mirror, and its image I.M is on the same side of the centre, they will be both in the same 
situation. If the object AFB be farther from the concave mirror SGV than its principal locus, and it 31, E, 
I. be the places of the several foci, to which the rays from A, F, B, (by Prop. L.) will converge, a 
distinct image of the object will appear upon a paper placed at 31, E, I ; and it the paper be taken 

away, and the eye be more remote from the mirror than 31 El, the rays will diverge trom these loci 

and become the last image. But, because the extreme perpendiculars ICH, 3ICK, in which (by Prop^ 
Lt III.) the points A and B will appear, cross each other at the centre C, between the object and 
image. A. the highest point of the object, will appear at I, the lowest point ol the image, and the 
reverse ; that is. the image with respect to the object, will be inverted, or they will be in contrary 
situations. The same may be shown in like manner with respect to the convex mirror. 

PROP. C\ . In spherical mirrors, concave or convex, the diameter of the object 
is to the diameter of the image, as the distance of the object from the centre, to 
the distance of the image from the centre, and also as the distance of the object from 
the surface to the distance of the image from the surface. 

If the eye is any where in the line FG, or that line produced, FG is the optic axis; whence the 
23. visible length of the object AB, and also of the image 131, is proportional (as appears from Prop. 
LXYIII.) to a subtense of the optic angle perpendicular to FG. The visible extensions or lengths of 
the object and of the image being then perpendicular to the same line FG, are parallel to one .another. 
Hence in all the cases, the angles ACB, 1CM, are equal, and also the angles CAB, Cl31. Therefore 
(El. VI. 4.) AB, the visible length of the object, is to 311, that of the image, as AC. the distance of 
the object from the centre, is to IC, the distance of the image. 

Again, since the object AFB consists of real radiants, and the image 3IEI of imaginary radiants when 
the rays diverge, and of foci when they converge, after reflection ; and since when they diverge, FG, the 
distance of the object from the surface, is (by Prop. LYI.) to EG, the distance of the image from the 
surface, as FC, the distance of the object from the centre, is to EC, the distance of the image from the 
centre : but (by El. Y.I. 2.) AB is to 311, as FC to EC ; therefore AB is likewise to 311, as FG to EG; 
that is. tbe diameter of the object is to that of the image, as the distance of the object from the sur¬ 
face to that of the image from the surface. 

PROP. CVI. If the eye is close to a convex or concave mirror, the apparent 
diameter of the object is equal to the apparent diameter of the image. 

If the eye is at G. the real diameters of the object AFB, and image 3IE1 are (by Prop. CV.) as 
their respective distances from the eye. Therefore (by Prop. LXXI.) their apparent diameters will 
subtend the equal angles AGB, IGM, and will be equal. 

PROP. CVII. If the eye is placed in the centre of a concave mirror, it can see 
nothing in the mirror but its own image. 

For the eye. in this situation, is in the place of its own image, and therefore rays will be reflected 
to it from every point of the surface. 

PROP. CVIII. If an object is nearer to a concave mirror than its principal focus, 
the image appears behind the mirror, farther from the mirror and larger than the 
object, erect, and distinct. 


OB' VISION. • 


139 


Chap. IV. 


The object AFB being nearer to the concave mirror than the principal focus, the rajs which Plate 7 
diverge from each point in the object before the mirror will diverge after reflection (by Prop. L.) 1 '8- 23 - 
less than before from the imaginary radiants M, E, I; whence the image formed bv them will (Prop. 

\ I.) be farther from the mirror than the object; and consequently (by Prop. CV.) it will be larger than 
the object. Because the object is nearer the mirror than its principal focus, it is likewise nearer than 
the centre ; whence (by Prop. CIV.) the image will be erect. Lastly, the image will be seen distinctly, 
because the rays from it diverge as from objects at a moderate distance. 

PROP. CIX. If an object touch a concave mirror, the image will touch it like¬ 
wise, and they will be equal. 

For the real radiants being close to the mirror, the imaginary radiants are so too; whence (by 
Prop. CV.) their diameters will be equal. 

PROP. CX. If an object is placed in the focus of a concave mirror, the image is 
at an infinite distance behind the miror, larger than the object, erect, and distinct. 

When the object AFB is in the principal focus of the concave mirror SGV, the image is at an infinite 
distance. It will (by Prop. CV.) be larger than the object, on account of its remoteness. It will be 
seen erect (by Prop. CIV.) because it is on the same side of the centre with the object. And since the 
rays of each beam are parallel, it will be seen as distinctly, as the naked eye sees very remote objects. 

PROP. CXI. If the object be farther from a concave mirror than its focus, and the 
eye be nearer than the place of the image, the object will appear confused, behind the 
mirror, erect, and magnified. 

The rays which diverge from A, F, B, in an object more remote from the concave mirror SGV, than Plate 7. 
its focus, will (by Prop. L. ) be collected, and on a surface of white paper form an inverted image. If Fig- 21 
the eye is any where between B and G, the rays from every radiant are converging when they come 
from the mirror to the eye, whence it will appear confused, because the eye is not accustomed to see 
rays in this state. The rays AH, AG, diverging from A, will, after reflection, converge toward I; but 
if the eye be nearer to the mirror than I, the reflected ray Gl will not cross its perpendicular HI in any 
place before the eye, since they are in a state of convergency ; consequently, the apparent place of this, 
or any other point of the image, will be indeterminate. It will be seen erect, because MEI,the inverted 
image of the object, will be drawn inverted on the retina. Lastly, because the rays of each beam 
converge after reflection, as HI, GI, they will appear to come, not from points, but from circular spots 
larger than the points of the object, the image will appear confused, and be larger than the object. 

PROP. CXII. If an object is farther from a concave mirror than its principal focus, 
and the eye is farther from the mirror than the place of the image, the image appears 
before the mirror, inverted, and distinct. 

Rays coming from M, E, I, an object at a greater distance than the principal focus from the mirror Plate 7. 
SGV, will (by Prop. L.) converge to AFB, and would paint an inverted image upon a surface of white Fig. 21. 
paper placed there. From thence they will diverge, (the paper being taken away ); whence to the eye 
placed any where beyond AFB, the image will appear inverted. And it will be seen distinctly, because 
the rays come to the eye diverging, as from an object at a moderate distance. 

Schol. 1 . The inverted images of objects may be represented in a dark room by a concave mirror 
which receives rays passing from external objects through a hole in a window-shutter, and collects 
them into a focus on a surface of white paper. 

Schol. £. A concave mirror, collecting the parallel rays of the sun into a focus, will act as a burning- 
glass. 

PROP. CXIII. When an object is placed before a convex mirror, its image appears 
behind the mirror, nearer the mirror and less than the object, distinct, and erect. 

If AFB be an object placed before the convex mirror SGV, the rays which before reflection diverge plate 7. 
from A, F, B, will (by Prop. LVII.) after reflection diverge from as many radiants 1, E, M, behind the Fig. 22. 
mirror, forming the image. And because the reflected rays (by Prop. LU.) diverge more than the in¬ 
cident ones, the image IM (by Prop. VI.) will be nearer the mirror than the real radiants or object AFB ; 
whence (by Prop. CV.) the image will be less than the object. And because the reflected rays come 
to the eve in a state of divergency, the image will be seen as distinctly as any visible object, seen by 


140 


OF OPTICS. 


Book VI. 


Plate 7. 
Fig. 22. 


Plate 7. 
Fig. 24. 


\ 


such diverging rays, at the same distance. Lastly, if the object AFB were at an infinite distance from 
the mirror, the rays proceeding from any point in it would fall parallel upon the mirror, and therefore 
would upon reflection form the image in the principal focus, that is, in the middle point between G and 
C, or on the same side of C with the object ; whence (by Prop. CIV.) it would be erect. At any finite 
distance of the object, the image, being still nearer the surface, must therefore be erect. 

PROP. CXIV. When either the eye or the object departs from a convex mirror, 
the apparent diameter of the image decreases. 

If the object AFB continue in its place, the image IM will (by Prop. LVI.) be always at the same 
distance from the mirror; and (by Prop. CV.) the real diameter will be invariable; consequently (by 
Prop. LXIX.) the apparent diameter of the image will be inversely as the distance of the eye. 

If the object AFB depart from the mirror, the ratio of FG, the distance of the object, to GE, that of 
the image, (El. V. 3.) will increase ; whence (by Prop. CV.) the ratio of the diameter of the object to 
ihat of the image will likewise increase; that is, the image will become less with respect to the object; 
but the eye remaining in the same place, the apparent diameter of the image will (by Prop. LXX.) be 
as its real diameter; consequently, the apparent diameter will decrease. 

Exp. The Propositions in this section may be confirmed, by placing an object before a Plane, Con¬ 
cave, or Convex Mirror, according to the terms of their respective propositions. 


CHAPTER. V. 

Of Colours. 

SECTION. I. 

OF THE DIFFERENT REFRANGIBILITY OF LIGHT. 

Def. XXV. Rays of light are differently refrangible , when at the same or equal 
angles of incidence, some are more turned out of the way than others. 

Def. XXVI. Rays are differently refexible , when some are more easily reflected 
than others. 

Def. XXVII* Light is called hoiiiogeneous , when all the rays are equally refrangi¬ 
ble ; and heterogeneous , when some rays are more refrangible than others. 

Def. XXVIII. The Colours of homogeneous rays are called primary or simple 
colours ; those of heterogeneous, secondary or mixed. 

PROP. CXV. The rays of the sun are not all equally refrangible ; and those rays 
which have a different degree of refrangibility, have likewise a different colour. 

If a beam of light SF from the sun pass into a dark room through F, a round hole in a window- 
shutter EG, and be received upon a w hite surface, a white round image will be seen. If a glass prism 
ABC be so placed as to receive the beam of light, the rays of this beam, from their refraction in passing 
through the prism, will be turned upward, and the refracted image PT will be oblong, having its breadth 
equal to the diameter of the circular picture O. If all the rays were equally refracted upward, it is 
manifest that such a refraction would not change the form of the picture. Since therefore the refracted 
image is oblong, it must be formed by rays differently refrangible, which fall with equal angles of obliquity 
upon BC, the first side of the prism, but are some of them, in refraction, turned more out of the way than 
others ; those rays which go to P, the upper part of the image, being most refrangible, and those which 
go to T, the lower part, being least refrangible. 

This oblong image is of different colours in different parts, the whole image being made up of rays 
of seven different colours, in the following order, beginning with those which are most refrangible ; 



Chap. V. 


OF COLOURS. 


141 


violet, indigo, blue, green, yellow, orange, red. This refracted picture consists of several round 
pictures so near each other, that each higher circle mixes in part with that below it, whence the 
colours near the the upper and lower edge of each circle are blended. The sides of these circles being 
very near to each other appear like right lines. 

Exp. 1. Observe the prismatic image formed by the refraction of the rays passing through a single 
prism. 

2. To separate the several colours as much as possible, make the hole F in the window-shutter Plate 7. 
very small, and collect the rays which pass through it, into a focus L, by a convex lens MN. Let the Fig. 27. 
rays which have passed through the lens be now received upon a prism placed near the lens; the rays 

will be refracted upward into an oblong image. 

3. To prove that the prismatic image is produced by the different refrangibility of the rays, and by Plate 7. 
no other cause, let a second prism DH be placed beyond the first abc , at right angles to it. The rays Fig. 25. 
passing through this second prism are refracted sideways ; those which were most refracted upward by 

the first prism, are most refracted sideways by the second; but, the rays not being spread in breadth, 
the image remains of the same form. 

PROP. CXVI. Those rays of light, which are most refrangible, are also most 
reflexible. 

If the beam of light passing through F fall upon a prism ABC, whose sides AC, AB, are equal, and Plate 7. 
the angle at A a right angle ; when the obliquity of these rays, as they are to pass out of the prism at F ‘g- 26 * 
its base BC, is less than 40 degrees (by Cor. 3. Prop. XIII.), the greater part of the beam will pass 
through, but some rays will be reflected at the surface BC. Those rays which pass through the base 
(by Prop. CXV.) form an oblong coloured image at HG, between the most refrangible ray MH, and 
the least refrangible ray MG. If the rays which are reflected from M are made to pass through another 
prism XYV, they will also form a faint oblong coloured image p t. Now, if the prism ACB be turned 
slowly round upon its axis in the direction ACB, the obliquity of the rays FM to the base BC, will con¬ 
tinually increase till all the rays will be reflected at M. Consequently, the image p t will become much 
brighter than before. And this total reflection will not be produced at once, but the most refrangible 
rays MH will be first entirely reflected; for the violet colour in HG will first disappear, and the same 
colour at p will first become brighter. In like manner, as the prism ABC is turned round, each different 
sort of ray will be reflected sooner, as it has a greater degree of refrangibility. Hence it appears, that 
the rays of the sun have different degrees of reflexibility, and that those which are most refrangible 
are also most reflexible. 

PROP. CXVII. Homogeneous light is refracted regularly without any dilatation of 
the rays. 

Exp. When the rays of any colour in the oblong image, as gree'n, are separated from the rest, in 
the manner described in Prop. CXV. if some of these rays are transmitted through a small hole in a thin 
board, and refracted by a prism placed on the other side, the image formed by these rays after refrac¬ 
tion will not be oblong, but circular. 

PROP. CXVIII. The confused appearance of objects seen through refracting bodies 
is owing to the different refrangibility of light. 

Exp. Small objects placed in a sun-beam and viewed through a prism, will be seen confusedly; but 
if they are placed in a beam of homogeneous light, separated by a prism, they will appear as distinct 
through a prism, as when viewed by the naked eye. 

Schol. 1. Although the 13th Proposition (in which it was shown that when a ray of the sun is passing 
out of one medium into another, the ratio of the sine of incidence to the refracted sine will not be 
changed by changing the obliquity of the incident ray) proceeds upon the supposition, that all rays are 
equally refrangible, and therefore is not exactly true; the demonstration is strictly applicable to any one 
sort of rays, as the red ones, which are equally refrangible. 

Schol. 2. Since, all other circumstances being equal, the same cause, namely, the passing of the rays 
out of one given medium into another, will turn the violet rays more out of the way than the red rays ; 
the attracting force which acts upon both being the same, it is probable that any single ray of the least 
refrangible sort contains a greater quantity of matter than any single ray of the most refrangible sort. 

PROP. CXIX. The colours of homogeneous light can neither be changed by re ¬ 
fraction nor reflection. 


142 


Ptate 7. 

Fig. 28. 


OF OPTICS. Book VI. 

Exp. 1 . Let a beam of homogeneous light pass through a round hole in a pasteboard, and then be 
refracted by a prism on the other side; the colour of the rays will remain the same. 

2. Red lead, viewed in homogenous red light, will be red, but if placed in green, or any other 
homogeneous light, it will take the colour of the rays which fall upon it. 

PROP. CXX. The whiteness of the sun ? s light arises from a clue mixture of all 
the primary colours. 

Exp. If the oblong picture PT fall upon the convex lens MN, the rays, which were separated at PT, 
will, by passing through the lens, he collected into a focus at G, and form a round image of the sun upon 
apiece of paper DE. This image, formed of all the primary sorts of rays, is white. That the whiteness 
ofthe image is owing to the due mixture of all the sorts of rays, appears from hence, that, if any of the 
colours be intercepted at the lens, the image loses its whiteness. The paper being removed from DE 
to d e, the rays, having crossed at G, will form the prismatic image t p , inverted, but distinct; from whence 
it appears, that the colours are not changed by being mixed at the focus. 

PROP. CXXI. The colours of all bodies are either the simple colours of homogene¬ 
ous light, or such compound colours as arise from a mixture of homogeneous light. 

Each sort of light having a peculiar colour of its own, which no refraction or reflection can alter, 
since bodies appear coloured only by reflected light, their colours can be no other than the colour ot 
some single homogeneous light, or of a mixture of different sorts of light. 

PROP. CXXII. Water, air, glass, or any other transparent substance, when drawn 
into thin plates, becomes coloured. 

Ext. 1. If a soap-bubble be blown up, and set under a glass that the motion of the air may not af¬ 
fect it, as the water glides down the sides and the top grows thinner, several colours will successively 
appear at the top, and spread themselves from thence in rings down the side of the bubble, till thev 
vanish in the same order in which they appeared. At last a black spot appears at the top, and spread* 
till the bubble bursts. 

2. If a piece of plane polished glass is placed upon the object glass of a long telescope, and the 
interval between them is filed up with water, as the glasses are pressed together the same colour* 
arise at the point of contact, and spread themselves in circular rings round that point in the same 
order as in the soap-bubble. 

3. A convex and concave lens, of nearly the same curvature, being pressed closely together, ex¬ 
hibit rings of colours about the point where they touch. Between the colours there are dark rings, 
and when the glasses are very much compressed, the central spot is dark. Sir I. Newton found the 
thickness of the air between the glasses, where the colours apeared, to be as 1, 3, 5, 7, &.c. and the 
thickness where the dark rings appeared, to be as 0, 2, 4, 6, 8, &,c. The coloured rings must have 
appeared from the reflection of the light; and the dark rings from the transmission of it. Consequently, 
the rays were transmitted when the thickness of the air was 0, 2, 4, 6, 8, Sic. and reflected at the 
thicknesses 1, 3, 5, 7, &c. Sir I. Newton, therefore, supposed, that every ray of light, in its passage 
through any refracting surface, is put into a certain state, which, in the progress of the ray, returns 
at equal intervals, and disposes the ray, at every return, to be easily transmitted through the next 
refracting surface, and between the returns, to be easily reflected by it. These he calls fits of easy 
transmission and reflection. See Schol. Prop. XLVI. 

4. Two pieces of plate glass wiped clean, and rubbed together, will soon adhere with a conside¬ 
rable force, and exhibit various ranges of colours. One of the most remarkable circumstances attend¬ 
ing this experiment, is the facility with which the colours may be removed, or even made to disappear, 
by heats too low to separate the glasses. A touch of the finger immediately causes the irregular 
rings of colours to contract toward the centre, in the part touched. 

Cor. 1 . From these experiments it appears plain, that the colours of bodies depend, in some de¬ 
gree, upon the thickness and density of the particles that compose them. 

Cor. 2. Hence, if the density, or size of the particles in the surface of a body be changed, the 
colour is likewise changed. 

Schol. 1 . When the thickness of the particles of a body is such, that one sort of colour is reflected, 
other colours will be transmitted, and therefore the body will appear of the first colour. And, in 
general, a less thickness is found to be nescessary to reflect the most refrangible rays, as violet and 
indigo, than those which are least refrangible, as red and orange. 

Schol. 2. Sir 1. Newton, from a great variety of experiments on light and colours, concludes that 


Chap. V. 


OF COLOURS. 


143 


every substance in nature is transparent, provided it be made sufficiently thin. Gold, when reduced 
into thin leaves, transmits a bluish-green light. If we suppose any body, therefore, as gold, for instance, 
to be divided into a vast number of plates, so thin as to be almost perfectly transparent; it is evident, 
that all, or the greater part of the rays will pass through the upper plates, and when they lose their 
force, will be reflected from the under ones. They will then have the same number of plates to pass 
through, which they had penetrated before ; and thus, according to the number of those plates, 
through which they are obliged to pass, the object appears of one colour or another, just as the rings 
of colour appeared in Exp. 3. according to the distance of the glasses, or the thickness of the plates 
of air between them. 

The philosopher who has of late years most distinguished himself on the subject of light and colours 
is Mr. Delaval, who, by a great variety of well conducted experiments, has shown that colours are 
exhibited, not by reflected, but by transmitted light. This he proved by covering coloured glass, and 
other transparent coloured media, on the further surface, with some substance perfectly opaque, 
when he found that they reflected no colour, but appeared perfectly black. 

He concludes, therefore, that, as the fibres of mineral and animal substances are found, when clear¬ 
ed of heterogeneous matters, to be perfectly white, the rays of light are reflected from these white 
particles, through the coloured media with which they are covered ; that these media serve to inter¬ 
cept and impede certain rays in their passage through them, while a free passge being left to others, 
they exhibit, according to these circumstances, different colours. 

Mr. Delaval instituted other experiments with coloured fluids put into phials of flint glass, in the 
form of a parallelopiped. The bottom, and three sides of each phial, were covered with a black var¬ 
nish, the neck and the front being left uncovered. On exposing them to the incident light, he found 
that from the parts of the phials which were covered, no light was reflected, but it was perfectly 
black, while the light transmitted through the uncoated parts of the phials, was of different colours. 
The same fluids spread thinly on a white ground, exhibited their proper colours ; the light, indeed,, 
being in this case reflected from a white ground, and transmitted through a coloured medium. 

From these, and many other experiments, Mr. Delaval concludes, (1.) That the colouring particles 
do not reflect any light. (2.) That a medium, such as Sir I. Newton describes, is diffused over the 
anterior and further surfaces of the plates, whereby objects are reflected equally and regularly, as in 
a mirror. 

Our author next considers the colouring particles themselves, unmixed with other media. For this 
purpose, he reduced several transparent coloured lipuors to a solid consistence by evaporation with a 
gentle heat, which does not injure the colouring matter; and in this state also the colouring particles 
reflect no light, but are perfectly black. 

To determine the principle on which opaque bodies appear coloured, it must be recollected, first, 
that all the coloured liquors appeared such only by transmitted light; and secondly, that these liquors 
spread thinly upon white ground, exhibited their respective colours ; he therefore concludes, that all 
coloured bodies, which are not transparent, consist of a substratum of some white subtance, which is 
thinly covered with the colouring particles. 

On extracting, by means of spirits of wine, the colouring matter from the leaves, wood, and other 
parts of vegetables, he found that the basis was a substance perfectly white. He also extracted the 
colouring matter from different animal substances, as flesh, feathers, &c. when the same conclusion was 
obtained. Flesh consists of fibrous vessels, containing blood, and is perfectly white when divested of 
blood by ablution, and the red colour proceeds from the light which is reflected from the white fibrous 
substance, through the red transparent covering formed by the blood. The result was the same from 
an examination of the mineral kingdom. 

Some portions of light are reflected from every surface of a body, or from every different medium 
into which it enters. Thus transparent bodies reduced to powder, and water in the shape of froth, 
appear white, which is no other than a copious reflection of light from all the surfaces of the minute 
parts, and from the air interposed between them. 

For a full investigation of this curious and interesting subject, the reader must be referred to the- 
Memoirs of the Manchester Society, vol. ii. 


144 


OF OPTICS, 


Book VI. 


SECTION II. 

Of the Rainbow . 

PROP. CXXIII. When the rays of the sun fall upon a drop of rain and enter into 
it, some of them, after one reflection and two refractions, may come to the eye of a 
spectator, whose back is toward the sun, and his face toward the drop. 

Plate 8. If fl ie sun shine upon XY, a drop of rain, in any lines SF, SD, SA, &c. the greater part of the rays 

Fig. 1. enter the drop, and passing on to the second surface, will be transmitted through the drop. But at PG 

in the second surface some few rays will be reflected, and proceed in some such lines as NR, NQ,; and 
coming out of the drop in the lines RV, QT, they may fall upon the eye of a spectator, placed in those 
lines with his face toward the drop. These rays are refracted when they enter the drop, reflected 
from the second surface, and again refracted when they come out of the drop. 

Def. XXIX. When rays of light reflected from a drop of rain come to the eye, 
those rays, which excite a perception of light, are called effectual. 

PROP. CXXIV. When rays of light come out of a drop of rain, they will not he 
effectual, unless they are parallel and contiguous. 

Most of the rays, which enter the drop between X and A, passing out of the hinder surface between 
P and G, only a few rays are reflected, and come out of the drop through the nearer surfaces between A 
and Y. None of these, only the rays which are parallel to one another will be effectual, because if 
they diverge, they will be so far asunder when they come to the eye, that only a very few of them can 
enter the pupil, and no perception of colours will be excited. Also unless several parallel rays he 
very near each other, the rays will be too few to create any perception. 

PROP. CXXV. When rays of light come out of a drop of rain after one reflection, 
those will be effectual which are reflected from the same point, and enter the drop 
near one another. 

Plate 8. Any rays AB, CD, when they have passed out of the air into a drop of water, will be refracted 

Fig. 2. toward the perpendiculars BL, DL, (by Prop. XI.) And as the ray AB falls farther from the axis than 
the ray CD, AB will be more refracted than CD ; so that these rays, though parallel to one another at 
their incidence, may describe the lines BE, DE, after refraction, and be both of them reflected from one 
and the same point E. Now all rays which are thus reflected from one and the same point, when th^y 
have described the lines EF, EG, and after reflection emerge at F and G, will be so refracted, when 
they pass out of the drop into the air, as to describe the lines FH, GI, parallel to one another. If these 
rays were to return from E in the lines EB, ED, and were to emerge at B and D, they would be re¬ 
fracted into the lines of their incidence BA, DC, (by Prop. XII.) But if these rays, instead of being 
returned in the lines EB, ED, are reflected from the same point E, in the lines EG, EF, the lines of 
reflection EG and EF, will be inclined both to one another and to the surface of the drop, just as much 
as the lines EB and ED are. First, EB and EG make just the same angle with the surface of the drop ; 
for the angle BEX, which EB makes with the surface of the drop, is the complement of incidence, and 
the angle GEY, which EG makes with the surface is the complement of reflection; and these two are 
ecpial to one another, by Prop. XLV. In the same manner we might prove that ED and EF make equal 
angles with the surface of the drop. Secondly, the angle BED is equal to the angle FEG, or the reflect¬ 
ed rays EG, EF, and the incident rays BE, DE, are equally inclined to each other. For the angle of 
incidence BEL is equal to the angle of reflection GEL, and the angle of incidence DEL is equal to the 
angle of reflection FEL, by Prop. XII. Consequently, the difference between the angles of incidence is 
equal to the difference between the angles of reflection, or BEL — DEL is equal to GEL — FEL, or BED. 
to GEF. Since therefore either the lines EG, EF, or the lines EB, ED, are equally inclined both to one 
another and to the surface of the drop, the rays will be refracted in the same manner, whether they were 
to return in the lines EB, ED, or are reflected in the lines EG, EF. But if they were to return in 
the lines EB, ED, the refraction, when they emerge at B and D, would make them parallel. Therefore 
if they are reflected from one and the same point E in the lines EG, EF, the refraction, when they emerge 
at G and F, will likewise make them parallel. 

Farther, in order to render the rays which emerge at F and G effectual, they must not only emerge 
in a direction parallel to each other, but must enter the drop nearly at the same place. 

Plate 8 . Let XY he a drop of rain, AG the axis or diameter of the drop, and SA a ray of light, that comes 

Fig-1. from the sun and enters tlje drop at the point A. This ray SA, because it is perpendicular to both the 


Chap. V. 


OF THE RAINBOW. 


145 


surfaces, will pass straight through the drop in the line AGH without being refracted, (by Prop. XlW) 

But any collateral rays that fall about SB, as they pass through the drop, will be made to converge to 
their axis, and passing out at N will meet the axis at H, (by Prop. XI.) Rays which fall farther from 
the axis than SB, such as those which fall about SC, will likewise be made to converge ; but then their 
focus will be nearer to the drop than H, (by Prop. XI.) Suppose therefore 1 to be the focus to which 
the rays that fall about SC will converge ; any ray SC, when it has described the line CO within the 
drop, and is tending to the focus I, will pass out of the drop at the point O. The rays, that fall upon 
the drop about SD more remote still from the axis, will converge to a focus still nearer than I, suppose 
at K, (by Prop. XXI. note.) These rays therefore go out of the drop at P. The rays, that fall still 
more remote from the axis, as SE, will converge to a focus nearer than K, as suppose at L; and the 
ray SE, when it has described the line EO within the drop, and is tending to L, will pass out at the 
point O. The rays that fall still more remote from the axis, will converge to a focus stiil nearer. Thus 
the ray SF will, after refraction, converge to a focus at M, which is nearer than L, and having describ¬ 
ed the line FN within the drop, it will pass out at the point N. Now here we may observe, that as 
any rays SB or SC foil farther above the axis SA, the points N, or O, where they pass out behind the 
drop will be farther above G, or, that as the incident ray rises from the axis SA, the arc GNO increas¬ 
es, till we come to some ray SD, which passes out of the drop at P, and this is the highest point where 
any ray, that falls upon the side AX, can pass out; for any rays SE, or SF, that fall higher than SD, 
will not pass out in any point above P, but at the points O, or N, which are below it. Consequently 
though the arc GNOP increases, whilst the distance of the incident ray from the axis SA increases, till 
we come to the ray SD ; yet afterward the higher the ray falls above the axis SA, this arc PONG will 
decrease. 

As there are many rays which pass out of the drop between G and P, so (by Prop. XL1II.) some few' 
rays will be reflected from thence ; and consequently, the several points between G and P, which are 
the points where some of the rays pass out of the drop, are likewise the points of reflection for the rest 
which do not pass out. Therefore in respect of those rays which are reflected, we may call GP the 
arc of reflection, and may say that this arc of reflection increases, as the distance of the incident ray 
from the axis SA. increases, till w r e come to the ray SD; the arc of reflection is GN for the ray SB, it 
is GO for the ray SC, and GP for the ray SD. But after this, as the distance of the incident ray from 
the axis SA increases, the arc of reflection decreases ; for OG, less than PG, is the arc of reflection for 
the ray SE, and NG is the arc of reflection for the ray SF. 

From hence it is obvious, that some one ray, which falls above SD, may be reflected from the same 
point with some other ray, which falls below SD. Thus, for instance, the ray SB will be reflected 
from the point N, and the ray SF will be reflected from the same point; and consequently, when the 
reflected rays NR, NQ, are refracted as they pass out of the drop at R and Q, they will be parallel, by 
what has been shown in the former part of this Prop. But since the intermediate rays which enter the 
drop between SF and SB, are not reflected from the same point N, these two rays alone will be paral¬ 
lel to one another when they come out of the drop, and the intermediate rays will not be parallel to 
them. And consequently these rays, RV, QT, though they are parallel, after they emerge at R and Q, 
will not be contiguous, and for that reason will not be effectual, (by Prop. CXX1V.) The ray SD is re¬ 
flected from P, which has been shown to be the limit of the arc of reflection ; such rays, as fall just 
above SD and just below SD, will be reflected from nearly the same point P, as appears from what has 
been already shown. These rays therefore will be parallel, because they are reflected from the same 
point P; and they will likewise be contiguous, because all of them enter the drop at one and the same 
place, very near to D. Consequently such rays, as enter the drop at D and are reflected from P, the 
limit of the arc of reflection, will be effectual (by Prop. CXXIV.), since when they emerge at the part 
of the drop between A and Y, they will be both parallel and contiguous. 

PROP. CXXVI. When rays which are effectual emerge from a drop of rain after 
one reflection and two refractions, those which are most refrangible will, at their emer¬ 
sion, make a less angle with the incident rays than those do which are least refran¬ 
gible; by which means, the rays of different colours will be separated from one 
another. 

Let FH, GI, be effectual violet rays emerging from the same drop at F, G ; and FN, GP, effectual Plate 8; 
red rays emerging from the same drop at the same points. The violet rays (by Prop. CXXIV.) are Fi S- 2 - 
parallel among themselves, because they are effectual; for the same reason the red rays are parallel 
among themselves ; but on account of the difference of refrangibility of the violet and red rays, the 
violef ray GI is not parallel to the red rav GP, but they diverge from the point G; and so of the rest. 

Both the violet ray GI and the red ray GP are refracted from the perpendicular LO, but (by Prop. 

19 


146 


Plate 8. 
Fig. 2. 


Plate 8. 
t jg. 3. 


OF OPTICS. Book VI. 

CXV.) GI more than GP ; whence the angle IGO is greater than the angle PGO. If the incident ray 
AB be continued in the direction ABK, and if IG and PG be continued backward till they meet AB in 
K and W, the angle IKA is that which the violet or most refrangible ray makes at its emersion with 
the incident ray, and PVVA that which the red or least refrangible ray makes with the same. And the 
angle IKA (El. I. 16.) is less than the exterior angle PVVA. The same may be proved concerning the 
rajs FH, FN, or any other rays which emerge respectively parallel to GI, and GP. But (by Prop. 
CXXIV.) all the effectual violet rays are parallel to GI, and all the effectual red rays are parallel to GP. 
Therefore the effectual violet rays at their emersion make a less angie with the incident rays than the 
effectual red rays. And universally the more refrangible rays, at their emersion, make a less angle 
with the incident rays, than those w hich are less refrangible. And since the effectual rays GI, GP, of 
different colours make different angles with the incident ray AK at their emersion, they will be separat¬ 
ed fora one another; so that if the eye were placed in the beam FGHI, it would receive only rays of one 
colour from the drop XY, and in FGN P only rays of another colour. 

Schol. The angle which the effectual red rays make with the incident rays is found to be 42° 2', 
that of the violet rays 40° 17’. 

Exp. Let a glass glass globe filled with water be exposed to the rays of the sun ; let the eye of the 
spectator be so situated, that the least refracted ray from the drop, coming to the eye, shall make an 
angle of about 42° with the line passing through the eye and the sun, the red rays only will be seen ; 
if the place of the eye be changed so as to enlarge this angle, the red will disappear; but if the angle 
be lessened, the colours of the more refrangible rays will appear. 

PROP. CXXVII. If a line is supposed to be drawn from the centre of the sun 
through the eye of the spectator, the angle which any effectual ray after two refrac¬ 
tions and one reflection makes with the incident ray, will be equal to the angle which 
it makes with that line. 

Let I be the place of the eye of the spectator; QT a line drawn from the centre of the sun through 
the eye ; and AB a ray coming from the centre of the sun. These two lines AB, QT, on account of 
the great distance of the sun, may be looked upon as parallel. Therefore (El. I. 29.) the alternate an¬ 
gles AKI, KIT, or GIT, are equal., 

PROP. CXXVIII. When the sun shines upon the drops of rain as they are falling, 
the rays which come from those drops to the eye of the spectator, after one reflection 
and two refractions, produce the innermost or primary rainbow. 

Let TFY be the innermost or primary rainbow, the outer part of which TFY is red, the inner part 
VDX violet, and the intermediate parts, reckoning from the red to the violet, orange, yellow', green, 
blue, indigo. Suppose the spectator’s eye at A ; and let AI be an imaginary line from the centre of the 
sun to the eye of the spectator. If a beam of light S coming from the sun falls upon any drop F, and 
the effectual rays which emerge at F make an angle FA1 of 42° 2' with the line Al, these rays (by 
Prop. CXXVI1.) make the same angle with the incident rays, and consequently are red. Hence the 
drop F will appear red; for all the other rays which emerge from F, and would be effectual if they fell 
upon the eye, being refracted more than the red rays, will pass above the eye. If another beam of 
light S falls upon the drop D, and the effectual rays emerging at II make an angle of 40° 17' with the 
incident rays, the drop D will be of a violet colour; for all the other rays which emerge from H, and 
wmuid be effectual if they came to the eye, being refracted less than the violet rays, will pass below the 
eye. The intermediate drops between F and D w ill for the same reasons be of the intermediate colours. 
And that which has been proved concerning the drops in the line FD, may be shown of any other set of 
drops in which the angles made by the emerging and incident rays are equal. Thus, wherever a drop 
of rain is placed, if the angle w'hieh the effectual rays make with AI is equal to the angle FAI, or is 
42° 2', any such drop will appear red. If FAI was turned round upon the line AI, so that one end of 
this line should always be at the eye, and the other at I opposite to the sun, in this revolution the drop 
F would describe a circle, of which 1 would be the centre, and TFY an arc. And since in this revolu¬ 
tion the angle FAI continues the same, if the sun were to i hine upon this drop as it revolves, the effectual 
rays (by Prop. CXXVII.) would make the same angle with the incident rays in whatever part of the 
arc TFY the drop may happen to be; and consequently in whatever part of the arc the drop F is it 
will appear red. Now as innumerable drops are filling at once in right lines from the cloud, whilst one 
drop is at F, there will be others at T, Y, and every other part of the arc, which will appear red in 
the same manner that F would have done in the supposed circular revolution. Therefore, when the 
sun shines upon the rain, there will be a red arc TFY produced opposite to the sun. In like manner 
a violet arc VDX will be produced, and other intermediate arcs of the several intermediate colours 
which will together make up the primary rainbow. 


Chap. V. 


OF THE RAINBOW. 


147 


Schol. Cascades and fountains, whose waters, in their fall, are divided into drops, will exhibit rain¬ 
bows to a spectator, properly situated, during the time of the sun’s shining. This appearance is also 
seen by moonlight, though seldom sufficiently vivid to render the colours distinguishable. Dr. Gregory, 
in his excellent Economy of Nature, says, that he once saw a lunar bow ; it was in autumn, the night 
was uncommonly light but showery, and the colours much more vivid than he could have conceived. 

There were not so many colours distinguishable as in the solar bow. Coloured bows have been seen 
on grass formed by the refraction of the sun’s rays in the morning dew. 

Artificial rainbows may be produced by candle light on the drops of water ejected by a small foun¬ 
tain, or jet (P eau , or from the stream emitted from an feolipile. But the most natural and pleasing is 
by means of the air fountain, the jet of which is perforated with a great number of very fine holes, from 
which the water spouts so as to form a kind of liuted column. The rainbow is formed by the sun’s rays ; 
for the spectator has only to place the spouting streams directly in the sun’s beams, with his own back 
to the sun, and being in a direct line with the sun and centre of the jet, by stooping his head to a cer¬ 
tain degree, he will discover the beautiful appearance of the natural prismatic colours, and a small rain¬ 
bow, on the same principle as those which are seen in the time of rain and sunshine. 

PROP. CXXIX. The primary rainbow is never a greater arc than a semicircle. 

Since the line A1 is drawn from the sun through the eye of the spectator, and through I the centre Plate 8. 
of the rainbow, this centre is always opposite to the sun. And since the angle FAI is an angle of 42° Fig. 3. 
2', F, the highest part of the bow, is 42° 2' from I, its centre. If therefore the sun is more than 42° 

2' above the horizon, I, which is opposite to it, must be more than 4i° 2' below the horizon, and no 
primary rainbow will be seen. As much as the altitude of the sun is less than 42° 2', so much will the 
highest point F of the rainbow be above the horizon; and when the sun is in the horizon, 1, the centre 
of the bow, will also be in the horizon on the opposite side, and half the circle will be visible ; but 
when the sun is set, no bow can be seen. ' # 

PROP. CXXX. When the rays of the sun fall upon a drop of rain, some of them 
after two reflections and two refractions may come to the eye of a spectator, who has 
his hack toward the sun and his face toward the drop. 

If parallel rays from the sun, ZV, YW, fall upon the lower part of the drop of rain BGW, they will Plate 8. 
be refracted toward the perpendicular VL, WL, in entering the drop, and proceed in the direction VH, Fig- 5 - 
WI. At HI some part of these rays will (by Prop. XLIII.) be reflected into the directions HF, IG. 

And some of these rays will be again reflected at F, G, into the directions FD, GB ; which rays, when 
they emerge out of the drop at B and D, will be refracted from the perpendiculars, and may come to 
the eye of a spectator whose back is toward the sun and his face toward the drop. 

PROP. CXXXI. Those rays which are parallel to one another after they have 
been once refracted and once reflected in a drop of rain, will he effectual when they 
emerge after two refractions and two reflections. 

The contiguous rays ZV, YW, being refracted toward the perpendiculars VL, WL, when they en-Plate s. 
ter the drop, will (by Prop. XVIII.) become convergent; and because these rays fall upon the drop Fig- 5- 
very obliquely, their focus will not be far from the surface VW. If this focus be at K, the rays, after 
they have passed the focus, will diverge from thence in the directions KH, KI; and if KI be the fo¬ 
cal distance of the concave reflecting surface HI, the reflected rays HF, IG, (by Prop. L.) will be 
parallel. These rays are reflected again from the concave surface FG, and will meet in a focus at E, 
so that GE will be the focal distance of this reflecting surface; and because III, FG, are parts of the 
same sphere, the focal distances GE, KI, are equal. When the rays have passed the focus E, they will 
diverge in the lines EB, ED. Now, if the rays VK, WK, when they have met at K, were to be turn¬ 
ed back in the directions KV, KW, on emerging at V and W, they would (by Prop. XX.) be refract¬ 
ed into the lines of incidence, and become parallel. But since GE is equal to IK, the rays ED, EB, 
which diverge from E, fall in the same manner upon the drop at D and B, as the rays KV, KW, 
would fall upon it at V and W, and ED, EB, have the same inclination to the refractfng surface DB, 
as KV, KW, would have to VW ; whence the rays ED, EB, emerging at D and B, wi 11 be refracted 
in the same manner, and will have the same situation with respect to one another, as KV, KW, would 
have, that is, will be parallel to one another; having been contiguous before their entrance into the 
drop, they will therefore (by Prop. CXXIV.) be effectual. 

PROP. CXXXII. When effectual rays emerge from a drop of rain after two re¬ 
flections and two refractions, those which are most refrangible will at their emersion 


148 


OF OPTICS. 


Book VI. 


Plate 8. 
Fig. 5 


Plate 8. 
* ig. 5. 


Plate 8. 
Fig. 4. 


Plate 8 
Fig. 3. 


make a greater angle with the incident rays than the least refrangible will make 
with them ; by which means the rays of different colours will be separated. 

Let BM, BA, a violet and a red ray, emerge from B ; the angle which the violet ray BM makes 
with the incident ray YW is YrM ; and that which the red ray BA makes with the same is YSA. And 
sine BSY, the external angle of the triangle Br S, is (El. I. 16.) greater than the internal angle BrS 
or B r Y ; YrM, the complement of B r S, is greater than YSA, the complement of BSY. Conse¬ 
quently, since the emerging rays make different angles with the same incident ray, the refraction 
which they suffer at emersion will separate them from one another. 

Schol. The angle which the violet rays make with the incident ones is found to be 54° 7', and 
that of the red rays 50° 57'. 

PROP. CXXXIII. If a line be supposed to be drawn from the centre of the sun 
through the eye of the spectator, the angle which, after two refractions and two re¬ 
flections, any effectual ray makes with the incident ray, will be equal to the angle 
which it makes with that line. 

If YW be an incident ray, and BA an effectual ray, and AO a line drawn from the centre of the sun 
through A, the eye of the spectator, YW and AO may be considered as parallel; whence the alter¬ 
nate angles YSA, SAO, (El. I. 29.) will be equal. 

PROP. CXXXIV. When the sun shines upon the drops of rain as they are fall¬ 
ing, the rays which come from those drops to the eye of the spectator, after two re¬ 
flections and two refractions, produce the outermost or secondary rainbow. 

When the sun shines upon a drop of l'ain E in the outer edge of the secondary rainbow CBD, the 
effectual violet ray EA (by Prop. CXXXII. Schol.) makes an angle EAI of 54° 7' with AI, a line drawn 
from the sun through the eye of the spectator, and therefore (by Prop. CXXXIII.) makes the same 
angle with the incident ray SB. Therefore if the spectator’s eye be at A, all the rays except the 
violet will (by Prop. X.) make a less angle with AI, than EA, and fall above the spectator’s eye. 
In like manner it may be shown, that from the drop F, only red rays will come to the spectator’s eye, 
the rest falling below it; and that the rays emerging from the intermediate drops between E and F, 
and coming to A, will emerge at intermediate angles, and present to the eye the intermediate colours. 
If EAI be conceived to turn round upon the line AI, in such a revolution of the drop E, the angle EAI 
would remain the same, and consequently the emerging rays would make the same angle with the inci¬ 
dent rays. But in such a revolution the drop E would describe a circle, of which I would be the centre, 
and CBD an arc. Consequently, since the emerging rays make the same angle with the incident ones 
when the drop is at any other part of the arc, as at E, the colour of the drop will be violet to an eye 
placed at A, in whatever part of the arc the drop is placed. Now, since there are innumerable drops 
of rain falling at once, while one drop is at E, there will be others in all parts of the arc, which will 
all appear violet-coloured, for the same reason that E would have appeared of this colour in any oth¬ 
er part of the arc. In like manner, as the drop F appears red at F, and at any part of the arc FD, 
so will any other falling drop when it comes to any part of that arc. The intermediate arcs are form¬ 
ed in the same manner with the violet arc CBD, and the red arc FD ; and thus the whole secondary 
rainbow is produced. 

PROP. CXXXV. The colours of the secondary rainbow are fainter than those 
of the primary, and are ranged in the contrary order. 

At every reflection many rays pass out of the drop without being reflected; consequently the sec¬ 
ondary rainbow, which is produced after two reflections, is formed by fewer rays than the first, which 
is produced after one reflection. 

Again, in the primary bow, the violet rays, whtn they emerge effectually, make a less angle with 
the incident rays (by Prop. CXXVI.) and therefore (by Prop. CXXVII.) with the line AI, than the red 
rays. But the rays are here only once reflected, and the angle which the effectual rays make with 
AI is the distance of the coloured drop from I, the centre of the bow. Therefore the violet arc in 
the primary bow will be nearer to the centre of the bow than the red arc ; that is, the innermost col¬ 
our will be violet, and the outermost red. But in the secondary rainbow, the rays are twice reflect¬ 
ed ; and (by Prop. CXXXII .) the violet rays, which emerge so as to be effectual after two reflections, 


Chap. VI. 


OF OPTICAL INSTRUMENTS. 


149 


make a greater angle with the incident rays, that is, with the line AI, than the red ones; which 
angle is the distance of the violet arc from I, the centre of the bow. Therefore the violet arc in the 
secondary bow will be farther from the centre of the bow than the red arc; that is, the outermost 
colour is violet and the innermost red. 

PROP. CXXXVI. The secondary rainbow is never a greater arc than a semicircle. 

This is proved in the same manner as Prop. CXXIX. with this difference, that, since the rays of the 
highest colour in the secondary bow make an angle of 54° 7' with AI ; this bow will begin to appear 
when the altitude of the sun is less than 51° T; and when the sun is in the horizon on one side, this bow 
will have its centre in the horizon on the other side at the distance of 54° 7' from its highest point. 


CHAPTER VI. 

Of Optical Instruments . 

SECTION I. 

Of 'Telescopes . 

Dei\ XXX. An Astronomical Telescope consists of two convex lenses, whose dis¬ 
tance from each other is equal to the sum of their principal foci; that lens which is to¬ 
ward the object, is called the object-glass ; that which is next the eye, is called the 
eye-glass . 

If NL is one convex lens, whose focal distance is MF, and BD another, whose focal distance is CF; p^ te 7 8, 
and if these are so placed that the distance between them is equal to MF added to CF, that is, MC, they 
form an astronomical telescope. 

PROP. CXXXVII. Very remote objects, seen through an astronomical telescope, 
appear distinct and inverted. 

Let PM, PL, PN, be rays coming (by Prop. VIII.) parallel from the middle point in a very distant Tlate^S. 
object; let AN, AM, AL, come from the lowest point, and Q,N, QM, QL, come from the highest point. F, S' 7 ‘ 
These parallel rays will (by Def. XVIII.) be collected into the focus, and there form an image of the 
object, which (by Prop. LXXXII.) forms the object of refracted vision. But, by the construction of the 
telescope, GFE is the focus of the eye-glass. Consequently the rays which diverge from any point G in 
this image will, (by Prop. XX.) after they have passed through the eye-glass, become parallel. There¬ 
fore if the eye is at any point on the other side of the eye-glass, the object of refracted vision may be 
«een as distinctly as any very remote object can be seen by the naked eye ; and because the image is 
the object of vision, (by Prop. XXV.) it will be seen inverted. 

PROP. CXXXVIII. The apparent diameter of an object, seen through an astro¬ 
nomical telescope, is to the apparent diameter of the same object seen by the naked 
eye at the station of the object-glass, as the distance of the image from the object-glass 
is to its distance from the eye-glass. 

If the imaov, formed by the object-glass NL, were received upon a paper at EFG, the apparent P i ate> g> 
diameter of the object seen by the naked eye at M, the station of the object-glass, would be (by Prop. Fig. 7. 
LXXXIV.) equal to the apparent diameter of the image seen from the same station. Now the real 
diameter of the image is given, because its distance MF from the lens is given. Consequently, the 
apparent diameter of the image (by Prop. LXIX.) will be inversely as the distance of the eye from it. 

It the eve be placed at C, the station of the eye-glass, and consequently its distance from the image be FC, 
vhe imao-e will appear to the eye in that station bigger than at the station M (by Prop. LXXXIX.) in 
-he inverse ratio of the distances FC, IMF; that is, the apparent magnitude of the image at C will be to 


■150 


OF OPTICS. 


Book VI. 


Plate 8/ 
*'ig- 7. 


Plate 8. 
Fig. 7. 


Plate 8. 
Fig. 7.j 




that at M, as MF to FC. But the apparent magnitude of the image seen from M is equal to that of the 
object seen by the naked eye. Therefore the image seen from C appears bigger than the object, in 
the ratio of MF to FC. This would still be the case (by Prop. LXXXIV.) if the eye-glass were placed 
between the eye and the image, touching the eye. And since the image is in the focus of the eye-glass, 
the apparent magnitude (by Prop. XC.) is the same, whether the eye is close to the lens, or at any 
distance from it. Therefore wherever the eye is, the apparent diameter of the object, seen with the 
telescope, is to the apparent diameter of the same object seen by the naked eye at the station of the 
object-glass, as MF to FC, or as the distance of the distinct image from the object-glass, to its distance 
from the eye-glass ; that is, as the focal distance of the object-glass is to the focal distance of the eye¬ 
glass; consequently if the former be divided by the latter, the quotient will express the magnifying 
power; thus, if MF: FC :: 10 : 1, the telescope will magnify 10 times in diameter. 

PROP. CXXXIX. A telescope will not magnify an object, unless the focal distance 
of the object-glass be greater than the focal distance of the eye-glass. 

The rays which come from distant objects being nearly parallel, the image GFE (by Def. XVIII.) 
will be in the focus of the object-glass, which by the construction of the telescope, is also the focus of 
the eye-glass. But the apparent diameter of an object seen through a telescope, is to its apparent 
diameter when seen by the naked eye, (by Prop. CXXXVIII.) as the distance of the image from the ob¬ 
ject-glass, to its distance from the eye-glass ; that is, by what has been just proved, as the focal distance 
of the object-glass, to the focal distance of the eye-glass. Consequently, if MF, the focal distance of the 
object-glass, is greater than FC, the focal distance of the eye-glass, the object will be magnified ; if 
MF be equal to FC, the object will appear as to the naked eye ; if MF be less than FC, the object will 
appear diminished. 

Cor. 1 . Hence the object-glass of a telescope should be less convex than the eye-glass. 

Cor. 2. An object will be equally magnified by two telescopes of very different lengths, if the ratio 
of the focal distances of the object-glass and eye-glass be the same in each. 

Cor. 3. If a telescope be inverted, objects seen through it will be diminished; for the object-glass, 
which has.the greater focal distance, then becomes the eye-glass. 

PROP. CXL. The visible area, or space which may be seen at one view through 
a telescope, is as the area of the eye-glass. 

If GFE is any image, its distance from the object-glass being equal to the focal distance of the lens, 
the area of the image (by Prop. XXXVI.) is given ; but the quantity of this image which can be seen 
at one view must be greater or less, according to the magnitude of the hole through which it is seen ; 
that is, must be as the area of the eye-glass. 

PROP. CXLI. The brightness of an object seen through a telescope depends 
upon the area of the object-glass, but not the visible area. 

The brightness of the image, that is, of the object of refracted vision, is (by Prop. XXXVIII.) as 
the area of the lens which forms it, that is, of the object-glass. But (by Prop. XXXVI. Schol. 2.) the 
magnitude of the image is the same, whether the area of the object-glass is great or small; and con¬ 
sequently, if we look at it through an eye-glass of a given area, the quantity to be seen at once will 
not be altered by any change in the area of the object-glass. 

PROP. CXLII. The distance of the eye from the eye-glass, should be equal to the 
principal focal distance of the eye-glass. 

Since the image GFE is in the focus of the lens DCB, wherever the eye is placed on the other 
side of the glass, the image will appear equally magnified. But when the eye is just as far from the 
eye-glass as its focal distance, the visible area will be the greatest; for, in that case, (by Def. XVIII.) 
none but rays parallel, before the refraction, to MC the axis of the telescope, and therefore to the 
sides of the cylindrical tube in which the lenses are placed, can reach the eye, and consequents, no 
rays can come from the inner surface of this tube to the eye to make it visible ; whereas in any other 
station of the eye, oblique »iys from that surface would make the sides of the tube visible; whence 
the area of the vision, which remains the same, being (by Prop. CXL.) always as the area of the 
eye-glass, will be in part occupied by the sides of the tube, and the object will be seen only through 
the remaining part. 


151 


Chap. VI. OF OPTICAL INSTRUMENTS. 

D ef. XXXI. A telescope, consisting of four convex lenses, is a double Astronom¬ 
ical Telescope . 

Let the two lenses NML, and B, placed at the distance MB, equal to the sum of their focal distan- p ]ate g 
ces, form one telescope, and two lenses, C, D, placed at the distance CD, equal to the sum of their Fig. 8. 
focal distances, form another. If these two telescopes are fixed at the distance CB from each 
other, so as to be both used together, they form a double telescope; the lens LMN next to the object, 
is called the object-glass, and the lens B next the object-glass is called the first eye-glass, C the sec¬ 
ond, and D next to the eye the third. 

PROP. CXLIII. An object seen through a double telescope appears distinct and 
erect. 

The parallel rays which fall upon the object-glass NML (by Prop. CXXXVII.) form a distinct in- Plate 8. 
verted image at GFE, the focus of the object-glass. This image being also in the focus of the first 8 - 
eye-glass B, the rays of each beam from the several points of this image will become parallel by pass¬ 
ing through B ; whence, falling parallel on the second eye-glass C, they will form a distinct inverted 
image at KIH, the focus of this second eye-glass; and because KIH is also the focus of the third eye¬ 
glass D, the rays from this image, after passing through this third .eye-glass, will come to the eye 
parallel to each other. Consequently, the object will be seen distinctly; and because the second image 
is inverted with respect to the first, which is inverted with respect to the object, the second image, 
or object of refracted vision, is in the same situation as the object itself. 

PROP. OXLIV. A double telescope magnifies an object in the ratio of the focal 
distance of the object-glass, to the focal distance of the first eye-glass. 

The first telescope MB magnifies the object in the ratio of MF to FB ; and the second telescope Plate 8. 
CD is commonly made up of two lenses of equal convexities, which will not alter the apparent mag- Fi §- 8 - 
nitude of the objects. Therefore, when both are used together, the object is only magnified by the 
first in the ratio of MF, the focal distance of the object-glass, to FB, the focal distance of the first 
eye-glass ; consequently, the magnifying power is found by dividing MF by FB. 

Schol. 1 . The different refrangibility of the rays of light makes refracting telescopes imperfect; 
for those rays which are most refracted by passing through the lens, wili be brought to a focus and 
form an image nearer to the object-glass than those which are less refracted; and consequently the 
several sorts of rays are not properly collected in one focus to produce a perfectly white image, but 
each has its own focus, producing a confused and coloured image. 

Of two refracting telescopes which magnify equally, the shorter will give a more imperfect image 
than the longer. For the image appearing equal in both, but being farther from the object-glass in 
the longer than the shorter, must be in reality larger or more magnified; whence the defect arising 
from the different refrangibility of the rays will be more visible in the longer than in the shorter teles¬ 
cope. Hence, reflecting telescopes are more perfect than refracting ones; for when all the rays 
are reflected, their angles of incidence and reflection being equal, they will all meet in a focus at the 
same distance. 

Schol. 2. To remedy the defect of refracting telescopes, arising from the different refrangibility 
of rays of light, a compound object-glass was invented by Mr. Dollond, consisting partly of white flint 
Hass, and partly of crown glass, which have different refracting powers. These refract contrary ways; 
and the excess of refraction in the crown glass is made such, as to destroy the colour caused by the 
flint glass. A telescope thus formed is called achromatic. Let ABED represent a double concave riate. 12. 
lens of white flint glass, and AGDF a double convex of crown glass; then the part of the lenses which Fi S- l0> 
are on the same side of the common axis, viz. ACB and AFG may be conceived to act like two prisms 
which refract contrary ways; and if the excess of refraction in the crown glass be such as precise¬ 
ly to destroy the divergency of colour caused by the flint glass, the incident ray SH will be refracted 
to X without any production of colour; the same is true of the ray s h, and of all the other incident 
rays; and consequently the whole focal image, formed by this compound object-glassr will be achro¬ 
matic, or free from colour which might arise from refraction. It will therefore bear a larger aperture, 
and o-reater magnifying power, and of course enlarge objects much more than a common refracting 
telescope of the same length. The great impediment to the construction of large achromatic teles¬ 
copes, is the want of a flint glass of an uniform density. Dr. Blair has, within these few years, dis¬ 
covered that certain fluids, particularly those which contain the muriatic acid, may be formed into 


OF OPTICS. 


Book VX 


lenses. With these he has produced achromatic telescopes, which seem as perfect as the thing’ will 
admit of. See Transactions of the Edinburgh Royal Society. Also, Encyc. Brit. Art. Telescope, 
Vol. XVIII. Part i. 

Schol. 3. The construction of the eye has excited a suspicion, that that might be an achromatic instru¬ 
ment; but as the successive refractions are all in the same direction, that notion cannot be maintained, 
unless one of the humours is found to refract the red rays more than the violet, or all the humours 
refract all the rays equally. 

PROP. CXLV. To explain the construction and use of several kinds of telescopes. 

I. Of Galileo’s Telescope. 

Galileo’s telescope consists of a convex object-glass and a concave eye-glass, so placed that the dis¬ 
tance between them is the difference of their focal distances. 

In this telescope ZYX, a convex lens, is placed at the distance from BA, a concave lens, of YC, the 
difference between YF, the focal distance of ZX, and CF, the focal distance of BA. 

From a distant object let rays fall upon the convex lens YZ, from which they will proceed toward 
the focus of this lens at FG. But the concave lens AB, the focus of w'hich is at FG, renders the con¬ 
verging rays parallel when they reach the eye ; whence an image will be formed upon the retina. And 
the pencils of rays being made more diverging by passing through the concave lens, the visible image 
is seen under a larger angle than the object, and appears magnified. Also, because the pencils which 
form the image only cross one another once, the image appears erect. 

II. Of Sir Isaac Newton’s Telescope. 

In the tube ABCD, toward the end BC, let the concave mirror GH be placed perpendicular to DC, 
the lower side of the tube. If an object, which is at such a distance that rays coming from the same 
point may be considered as parallel to one another, be placed before the open end of the tube AD, these 
parallel rays will be reflected from the concave mirror GH, and becoming convergent, would (by Prop. 
CXII.) form an inverted picture of the object upon a paper held at the focus of the mirror. But if the 
converging rays, before they reach the focus, fall upon a plain mirror K, placed at an angle of 45 degrees 
with D & C, the side of the tube, or with the axis of the telescope, they will be reflected from thence, and 
meet before it at L, forming an image perpendicular to the object, or parallel to the axis of the telescope. 
If this image be placed in the focus of the convex lens L, fixed in the side of the telescope, the eye 
will see it distinctly through the lens. 

The image seen from the station of the eye-glass L, either with or without the glass, will (as in the 
refracting telescope, see Prop. CXXXVIII.) appear as much larger than when seen from the concave 
mirror, that is, as much larger than to the naked eye, as the distance of the image from the eye-glass 
is less than its distance from the mirror, or as its distance from the mirror is greater than its distance 
from the lens. 

III. Of Gregory’s Telescope. 

In the tube TTYY, let a concave mirror EA be placed. Any parallel rays 00, PP, form an object 
A. falling upon this mirror, will, after reflection, (by Prop. CXI.) form an inverted image at C, its focus. 
Let C be more remote from a second smaller concave mirror PO (placed parallel and opposite to the 
first mirror EA in such manner that their axes shall be in the same straight line) than its focus. The 
rays which diverge from the several points of the image at C, and fall upon the mirror PO, will (by 
Prop. L.) converge after reflection; and consequently, if they pass through a hole NM in the first mir¬ 
ror EA, they will form a second image, w hich will be inverted in respect of the first, and in the same 
position’with the object. If, whilst these rays are converging, they pass through a plano-convex lens 
S P o (placed in a smaller tube joined to the larger), they will be brought to a focus sooner than they 
would' otherwise have been, forming the second image F. This erect image is seen by the eye at O, 
through a meniscal eye-glass LL, whose convexity is greater than its concavity. For the magnifying 
power of this telescope, see Musschenbroek. lntrod. ad Phil. Nat. or Priestley’s Optics, page 376. 

Schol. 1. In the telescopes made by Dr. Herschel, the object is reflected by a mirror as in the 
Gregorian telescope, and the rays are intercepted by a lens at a proper distance, so that the observer has 
his back to the object, and looks through the lens at the mirror. The magnifying power will be the 
same as in the Newtonian telescope; but there being no second reflector, the brightness of the object 
viewed in the Herschel telescope, is greater than that in the Newtonian telescope. 

The tube of Dr. Herschel’s grand telescope is 39 feet 4 inches in length, 4 feet 10 inches in diame- 


Chap. VI. 


OF OPTICAL INSTRUMENTS. 


153 


ter, every part of which is made of iron. The concave surface of the great mirror is 48 inches of 
polished surface in diameter, its thickness is 3| inches, and its weight is upward of 20001b. This noble 
instrument was, in all its parts, constructed under the sole direction of Dr. Herschel; it was begun in 
the year 1785, and completed August 28, 1789, on which day was discovered the sixth satellite ot 
Saturn. It magnifies 6000 times. 

Sciiol. 2. Dr. Priestley observes, that the easiest method of finding the magnifying power of any 
telescope, by experiment, is to measure the diameter of the aperture of the object glass, and that of the 
little image of it which is formed at the place of the eye. For the proportion between these gives the 
ratio of the magnifying power, provided no part of the original pencil be intercepted by the bad con¬ 
traction of the telescope. For, in all cases, the magnifying power of telescopes or microscopes, is 
measured by the proportion of the original pencil, to that of the pencil which enters the eye. Another 
method,' is to observe at what distance you can read any book with the naked eye; and then removing 
the book to the farthest distance at which you can distinctly read it by the help of the telescope. The 
book chosen for this purpose should be such, that the connexion of the subject should not assist the 
observer; as tables of logarithms, &c. Much depends on the steadiness with which the instrument is 
fixed. 

SECTION II. 


Of Microscopes. 


Def. XXXII. A single Microscope is one convex lens placed between a small object Plate s. 
and the eye. F,§ ' 6 ' 


Def. XXXIII. A double Microscope consists of two convex lenses, of which the 
object-glass is more convex than the eye-glass ; and the distance between them is equal 
to the distance of the image from the object-glass, added to the focal distance of the 
eye-glass. 


Let AB, a convex lens, be the object-glass, and EF, another convex lens, be the eye-glass. Let the 
small object KL be farther from the object-glass than its focus; an image MDN of the object will (by 
Prop. XXIV.) be formed behind the glass; let the distance of this image from the object-glass be ID, and s 
let its distance from the eye-glass be equal to the focal distance of the ej'e-glass; the distance of the 
two glasses from each other will be ID DX, or IX, that is, the distance of the image from the object- 
glass, added to the focal distance of the eye-glass. 

Some compound microscopes are made with three glasses, so that the rays after passing through AB 
the object-glass, and EF the eye-glass, are again made converging by a second eye-glass, and therefore 
brought sooner to a focus, than by the first, and the field of vision will be much greater than if only one 
lens were used. 

Con. Hence it appears, that the difference between the microscope and telescope is, that in the 
telescope the rays of each pencil fall upon the object-glass nearly parallel, and are united in its focus ; 
but in the microscope they fall upon it very much diverging from one another, and therefore form the 
image in a place beyond the focus, and consequently larger than the object. 


PROP. CXLVI. An object seen through a double microscope appears distinct and 
inverted. 


The pencils of rays issuing from the objects KL, being transmitted through the object-lens AB, their 
foci will be in MN ; where there will be an inverted image of the object, which is viewed through £!* te ® 
another lens, or eye-glass EF, the focus of which is at MN ; hence a distinct and direct image is formed * lg ‘ 
upon the retina , and it is seen inverted. 

PROP. CXLVII. The apparent diameter of an object seen through a double 
microscope is to that of the same object seen by the naked eye at the limit of distinct 
vision, in the compound ratio of the distance of the image from the object-glass, to its 


20 


05 C l 


154 


OF OPTICS. 


Book VI. 


distance from the eye-glass, and of the limit of distinct vision to the distance of the 
object-glass from the the object. 

The first part of this Proposition is demonstrated as Prop. CXXXVIII. And it is manifest (from 
Prop. LXIX.),that if Cl, the distance of the object-glass from the object, be less than the limit of distinct 
vision, the apparent diameter of the object will be as much greater than that of the object at the dis¬ 
tance at which the naked eye can see it distinctly, as IC is less than that distance. Therefore the object 
is magnified, because the distance of the image from the object-glass is greater than its distance from the 
eye-glass, and also because the distance from the object is less than the limit of distinct vision. The 
magnifying power of the microscope is then in the ratio compounded of these two ratios. Suppose ID 
=6IC ; and the eye-glass EF to be one inch focus, and the limit of distinct vision to be seven inches, 
then the diameter of the object KL will be magnified 6 X 1 =42; consequently 1764 times in surface. 

PROP. CXLVI1I. When the same eye-glass is used, the magnifying power of the 
microscope will be increased by increasing the convexity of the object-glass. 

For in order to keep the image in the focus of the eye-glass, when the convexity of the object-glass 
is increased, the object-glass must be brought nearer to the object ; the consequence of which will be, 
that the ratio of the limit of distinct vision to the distance of the object-glass from the object will (El. V. 
8.) be increased ; whence (by Prop. CXLVII.) the ratio of the apparent diameter of the object of re¬ 
fracted vision to that of the object seen by the naked eye will also be increased. 

. Schol. The aperture of the object-glass in a microscope must be small, else the outermost rays, 

diverging too much, will hinder the distinctness of vision ; but, on account of the smallness of the aperture, 
the object will appear faint, and it will appear necessary, in order to remedy this, to illuminate the 
object as much as possible. 

PROP. CXLIX. To describe the construction and use of the Solar Microscope. 

In a dark room, let a round hole be made in a window-shutter about three inches in diameter, through 
which the sun may cast a cylinder of rays into the room. In this hole let a tube be fixed, containing a 
convex lens of about two inches in diameter, and three inches focal distance; the object, placed be¬ 
tween two concave glasses, at the distance of about two inches and a half from the first convex lens ; 
and a second convex lens, whose focal distance is a quarter of an inch, placed at this distance from the 
object. Let a plane mirror, connected with the tube, and moveable by means of a wheel, receive the 
sun’s rays on the outside of the shutter, and convey them into the tube. The rays, passing through the 
first lens, will strongly illuminate the object, from which they will pass through the second lens, and 
form an inverted image of the object, magnified in the ratio of the distance of the object from the lens 
to that of the image from the lens, that is, in this case supposing the distance of the distinct picture to 
be twelve feet, or 144 inches, in the ratio of ^ : 144. Consequently, the diameter of the object will be 
magnified 576 times. 

SECTION. III. 

Of the Magic Lantern . 

PROP. CL. To describe the construction of the Magic Lantern. 

Plate 8 % In the side of a lantern let a tube be inserted, consisting of two parts, one moveable upon the other. 

Fig. J3. i n the moveable part let a convex lens GG be fixed ; in the immoveable part let an object EE, painted 

with transparent colours upon a piece of thin glass, be placed; and in the fixed part of the tube, a con¬ 
vex lens DD. This lens will cast a strong light from the candle upon the object EE. And when the 
rays which diverge from the several points of the object are, by the lens GG, made to converge, they 
will (by Prop. XXV.) form an inverted image of the object at KL, upon any white surface ; provided that 
the object is farther from the lens than its focus, and that the whole apparatus is placed in a dark room. 
The image KL will be larger than the object EE, in proportion as the distance of the image from the 
lens is greater than the object.* A concave reflector AB may be placed within the lantern, behind the 
candle, to increase the illumination of the picture EE. If the object be placed in an inverted position. 
Us image will appear erect 


Chap. VI. 


OF OPTICAL INSTRUMENTS. 


155 


SECTION. IV. 

Of the Camera Obscura. 

PROP. CLI. To describe the construction and use of the Camera Obscura. 

Let CD be a convex lens, and HK a plane mirror inclined at an angle of 45 degrees. An inverted Plate s. 
image of the object AB would be formed at EG, where the foci of the rays from the objects are found F 'S- l 4 * 
after refraction; but the rays being intercepted by the plane mirror HK, are reflected (by Prop. C.) to 
MM the focal distance before it, making an angle with the mirror of 45 degrees; whence the image 
will be in a position perpendicular to the object, at the top of the box, where, if the rays be received on 
a sheet of oiled paper, or a plate of glass unpolished on one side, it will be distinctly visible. 









/ 

















BOOK YII. 


OF ASTRONOMY. 


PART I. 

OF THE MOTIONS OF THE HEAVENLY BODIES. 


CHAPTER I. 

Of the Solar System in General. 

Def. I. ThE Solar System consists of the Sun, which is a luminous body ; seven 
primary planets, Mercury, Venus, the Earth, Mars, Jupiter, Saturn, and the Herschel: 
eighteen secondary planets, the Earth’s Moon, Jupiter’s four Satellites, Saturn’s seven, 
and six belonging to the Herschel; and an uncertain number of comets ; all which are 
opaque. 

Schol. Upon entering the subject of Astronomy, it will be proper briefly to describe the different 
systems which have been invented, in order to solve the natural appearances of the heavenly motions 

Ptolemy supposed the earth to be perfectly at rest, and the sun, moon, planets, comets and 
fixed stars to revolve about it every day; but that besides this diurnal motion, the sun, moon, planets 
and comets, had a motion in respect to the fixed stars, and were situated in respect to the earth in the 
following order; the Moon, Mercury, Venus, the Sun, Mars, Jupiter, Saturn. The revolutions of these 
bodies he supposed to be made in circles about the earth placed a little out of the centre. This system 
will not solve the phases of Venus and Mercury, and therefore cannot be true. 

The system received by the Egyptians was this; the earth was supposed immoveable in the centre 
about which revolved, in order, tbe Moon, Sun, Mars, Jupiter, and Saturn ; and about the Sun revolved 
Mercury and Venus. This disposition will account for the phases of Mercury and Venus, but not for the 
apparent motions of Mars, Jupiter, and Saturn. 

Another system was that of Tycho Brahe, a Danish nobleman, who was anxious to reconcile the 
appearances of nature with some passages of the Scriptures, taken in their literal interpretation. In 
his system, the earth is placed immoveable in the centre of the orbits of the sun and moon, without any 
rotation about its axis; but he made the sun the centre of the orbits of the other planets, which, there¬ 
fore, revolved with the sun about the earth. Objections to this system are, the want of that simplicity 
by which all the apparent motions may be solved; and the necessity of supposing that all the heavenly 
bodies revolve about the earth every day; also to suppose that a body should revolve in a circle about 
its centre without any central body is physically impossible. 

Some of Tycho’s followers, seeing the absurdity of a diurnal revolution of the heavenly bodies about 
the earth, gave a rotatory motion to the earth, and this was called the Semi-Tychonic system. 

The system, which is now universally received, is called the Copernican. It was formerly taught 
by Pythagoras, 500 years before Christ; and afterward rejected, till revived by Copernicus in the six¬ 
teenth century. Here the sun is placed in the centre of the system, about which the planets revolve 
from west to east, in the following order; Mercury, Venus, the Earth, Mars, Jupiter, Saturn and the 
Herschel planet; beyond which, at immense distances, are placed the fixed stars. The moon revolves 
round the earth ; and the earth turns about an axis. The other secondary planets move round their 
respective primaries from west to east at different distances, and in different periodical times. 





Chap. II. 


OF THE EARTH. 


157 


According to this doctrine, the Sun S is the centre of the system; Mercury a, Venus b , the Earth f, p] ate 9 
Mars e, Jupiter/, and Saturn h, revolve in elliptical orbits round the sun ; the moon d, revolves about Fig. 1 . 
the earth, and the satellites of Jupiter, Saturn, and the Herschel, revolve about their primaries ; and 
the planes of their orbits are inclined to one another.* 

This doctrine, being admitted as true, will account for the apparent motions, and other phenomena, of 
the heavenly bodies, as will be seen in the following Chapters. 


CHAPTER II. 

Of the Eart h. 

SECTION I. 

Of the Globular Form of the Earth, and its diurnal Motion about its Axis, and of the 

Appearances which arise from these. 

PROPOSITION I. 

The earth is of a globular form. 

For, 1. The shadow of the earth projected on the moon in an eclipse is always circular; which 
appearance could only be produced by a spherical body. 2. The convexity of the surface of the sea is 
visible ; the mast of an approaching ship beiug seen before its hull. 3. The north pole becomes more ele¬ 
vated by travelling northward, in proportion to the space passed over. 4. Navigators have sailed round 
the earth, and by steering their course continually westward, arrived, at length, at the place from whence 
they departed. 

Def. II. The Axis of the earth is an imaginary line passing through the centre, 
about which its diurnal revolution is performed. 

Def. III. The Poles of the earth are the extremities of this axis. 

Def. IV. The Equator is the circumference of an imaginary great circle passing 
through the centre of the earth, perpendicular to the axis, and at equal distances from 
the poles. 

Def. V. If the axis of the earth be produced both ways, as far as the concave sur¬ 
face of the heavens, in which all the heavenly bodies appear to be placed, it is then 
called the Axis of the Heavens ; its extremities are called the Poles of the Heavens ; 
and the circumference produced by extending the plane of the equator to the same 
concave surface, is called the Equator in the Heavens. 

Def. VI. Circles drawn through the poles of the earth or heavens, perpendicular 
to the plane of the equator, are called Secondaries of the equator. 

Def. VII. The sensible Horizon is an imaginary circle, which, touching the surface 
of the earth, separates the visible part of the heavens from the invisible. The ration¬ 
al Horizon is a circle parallel to the former, the plane of which passes through the 
centre of the earth. 

* Dr. Herschel, on the 13th of March, 1781, while pursuing a plan which he had formed of observing, with telescopes 
of his own construction, eveiy part of the heavens, discovered, in the neighbourhood of H. Geminorum, a planet far 
beyond the orbit of Saturn, -which had never before been visible to mortal eyes. He has since discovered six secondaries 
belonging to this new planet, which planet is called either the Herschf.l, from the name of its indefatigable and truly great 
discoverer; or the Georgium Sidus, or Georgian Planet, in honour of the late king, who distinguished himself astlie patron 
of Dr. Herschel. The planet is denoted by this character ; an H, as the initial of the name, the horizontal bar being 
crossed by a perpendicular line, forming a kind of cross, the emblem of Christianity, denoting, perhaps, that its discovery 
was made in the Christian era, as all the other planets were known long before that period. 



158 


OF ASTRONOMY. 


Book VII. Part L 


Plate 9., 
Fig. 4. , 


S'chol. Since (by Book VI. Prop. LXIX.) the apparent diameter of an object is inversely as its dis¬ 
tance, if the distance be increased in such manner that it may be looked upon as infinite, the apparent 
magnitude becomes a point. Hence AF, the semidiameter of the earth, viewed at the different distan¬ 
ces o, O, R, diminishes, till at the distance of O, a fixed star, it becomes’a point, and the star appears 
in the same place ki the heavens, whether viewed from the visible horizon SET, or rational horizon 
HBR. 

Def. VIII. The Poles of the Horizon are two points, the one of which, over the 
head of the spectator, is called the Zenith ; the other, which is under his feet, is called 
the Nadir, 

Def. IX. Circles drawn through the zenith and nadir of any place, cutting the hor¬ 
izon at right angles, are called Vertical Circles. 

Def. X. A vertical circle passing through the poles of the heavens, is a Meridian , 
and is said to be the meridian of any place through which it passes. 

Def. XI. The meridian of any place passing through Jthe poles, and falling per¬ 
pendicularly upon the horizon, cuts it in two opposite cardinal points, called North 
and South. 

Def, XII. A Meridian Line is the common intersection of the plane of the merid¬ 
ian and the plane of the horizon. 

Cor. Hence any line which lies due north and south in a horizontal plane, may be considered as 
part of the meridian line. 

Schol. 1 . To draw a meridian line; perpendicular to a horizontal plane, erect a wire, or stile, 
seven or eight inches long, and as it is not easy to determine precisely the extremity of the shadow, 
it will be best to make the stile flat at top, and to drill a small hole through it, noting the lucid 
point projected by it; mark, at several different times before noon, these lucid points, and through 
them draw concentric circles about the middle point of the wire’s station ; observe in the afternoon when 
the lucid points again touch these circles ; and find the middle point of each arc between the points 
already taken ; a line drawn through these middle points, and the common centre, will be the me¬ 
ridian line ; for, since at equal distances from noon, the sun is at the same height, or in verticals 
equally distant from the meridian, the circle drawn through the zenith at equal distances from these 
verticals is the meridian. This should be done about the summer solstice, between the hours of 
9 and 11 in the morning and 1 and 3 in the afternoon. 

Schol. 2. To observe the transit of any heavenly body over the plane of the meridian; place 
in this plane a telescope, having two cross hairs before its object-glass, one vertical, the other hori¬ 
zontal, and observe when the vertical hair passes through the centre of the heavenly body; or hanging 
two plumb-lines exactly over the meridian line, place your eye close to one of the threads in such 
manner, as that it shall cover the other thread, and observe when the body is behind the threads. 

Def. XIII. The Altiude or Depression of any heavenly body above or below the 
horizon, is the arc of a vertical circle intercepted between the body and the horizon, 
or the angle at the centre measured by that arc. 

Schol. The altitude of any heavenly body is found by the help of a quadrant thus ; bring the quad¬ 
rant into such a situation that the star may be seen through the sights ; then the angle, contained be¬ 
tween the string of the plummet and the side of the quadrant, on which the sights are not placed, is the 
altitude of the star. 

Def. XIV. The Prime Vertical is that which crosses the meridian at right 
angles in the zenith and nadir, cutting the horizon in the cardinal points East and 
West. 

Def. XV. The Azimuth of a heavenly body, is the arc of the horizon intercepted 
between the meridian and vertical circle passing through that body ; it is eastern or 
western, as the body is east or west of the meridian. 


Chap. II. 


OF THE EARTH, 


159 


Schol. The azimuth of any star may be thus found. Let AC be a given meridian line. Above P|ate 9. 
any point A in this line, let a cord with a plummet be hung; let another cord with a plummet be Fi S- 
hung at E, so that the star and the two cords shall lie in one and the same right line. Let the per¬ 
pendiculars AD, BE, represent the cords, and draw AB. From the point B to any point C, in the 
meridian line AC, taken at pleasure, draw the right line BC ; then with a scale of equal parts measure 
the three lines AB, AC, BC. In the triangle, therefore, ABC, there will given all the sides, from 
whence will be found the angle BAC, equal to the azimuth required. 

For if the meridian line be supposed to be continued to F, and the line BA to G, the angle FAG 
will be the azimuth of the star; but the angle FAG will be equal to the angle at the vertex BAC; 
therefore the angle BAC will be equal to the azimuth. 

Def. XVI. The Amplitude of a heavenly body at its rising is the arc of the hori¬ 
zon intercepted between the point where the body rises, and the east; its amplitude 
at setting is the arc of the horizon intercepted between the point where the body 
sets and the west; it is northern, or southern, as the body rises, or sets, to the north 
or south of east or west. 

Def. XVII. If a heavenly body rises, or sets, when the sun rises, it is said to rise 
or set cosmically ; if it rises, or sets, when the sun sets, it is said to rise or set achron- 
ically ; it is said to set or rise heliacally , when it approaches so near the sun as to be¬ 
come invisible, or recedes so far from him as to become visible. 

Def. XVIII. The Latitude of a place upon the surface of the earth is its distance 
from the earth's equator; it is measured by the arc of the geographical meridian of 
the place intercepted between the place and the equator : latitude is either northern 
or southern. 

Def. XIX. Parallels of Latitude are circles on the surface of the earth, drawn 
parallel to the equator. 

PROP. II. A degree in the equator is to a degree in any parallel of latitude 
as radius to the cosine of latitude. 

Let EPQ be a geographical meridian, EQ the equator, and FB a parallel of latitude. The cir- Plate 9. 
cumference EQ, is to the circumference FB, and any part of EQ, to any similar part of FB, as CQ, or CB F 'g- 3. 
the radius of EQ, to AB the radius of FB; and AB is the cosine of the arc BQ, which is the latitude of 
the parallel FB. Therefore a degree in EQ is to a degree in FB, as radius to the cosine of latitude. 

Def. XX. The Longitude of a place is the distance between the meridian of that 
place, aud the meridian of some other place, taken at pleasure, and called the first 
meridian ; it is measured by the arc in the equator intercepted between these two 
meridians. Longitude is either eastern or western, and is measured 180 degrees each 
way. 

PROP. III. The altitude of one pole, and the depression of the other, at any 
place on the earth’s surface, is equal to the latitude of that place. 

Let R be a place on the earth’s surface ; Z, N, its zenith and nadir; P, S, the poles of the heavens, Plate 9 
and F, s, the poles of the earth ; EE, the celestial equator, ce, the terrestrial, and HO the horizon. The Fi g- 2 - 
latitude of the place is e R, or the equal arc EZ ; and PO is the elevation of one pole, and HS the de¬ 
pression of the other. Because ZO is the distance of the zenith from the horizon, it is an arc of 90 
degrees; and because EP is the distance of the pole from the equator, it is also an arc of 90 degrees; 

ZO and EP are therefore equal. Take from each of these the common arc ZP, and the remainders 
EZ and PO are equal. But HS and PO are equal, because they subtend the equal angles HTS, PTO ; 
therefore the elevation of one pole PO, and the depression of the other IIS, are equal to the latitude 
of the place EZ. 

Cor. Hence the circumference of the earth may be measured, by measuring the length on the 
surface of the earth passed over in a line which lies north and south, while the pole gains one degree 
of elevation, and multiplying this length by 360. A degree of latitude contains 69^ English miles; 
whence 24930 miles is the measure of the circumference of the earth, and the radius 3956 ; but, as 
will be shown hereafter, the earth is a spheroid, whose polar diameter is to the equatorial, as 311 to 312. 


lGO 


OF ASTRONOMY. Book VII. Pakt I. 


Plate 9. 
Fig. 2. 





PROP. IV. The elevation of the equator at any place is equal to the complement 
of its latitude. 

Becauso ZO is equal to EP (each being an arc of 90 degrees) EZ is equal to PO, that is, (by Prop. 
Ill.) to the latitude of the place. But EH, the elevation of the equator, is the complement of EZ ; it 
is therefore equal to the complement of the latitude of the place. 

PROP. V. The earth revolving daily round its axis from west to east, the heaven¬ 
ly bodies will appear to a spectator on the earth to revolve in the same time from east 
to west. 

Let RCBF be the earth, T its centre, HTO the rational horizon to a spectator at R, whose zenith 
is Z ; let a star appear in the horizon at H. The earth revolving from west to east, that is, in the 
order of the letters, R, C, B, F, in a fourth part of one revolution, the spectator will be carried from 
R to C ; consequently, his horizon will become ZN, and the star which appeared in his horizon at H, 
when he was at R, will now appear nearly in the zenith. When another fourth part of the revolu¬ 
tion is completed, the spectator will be at B, and N being now his zenith, and HO his horizon, the star 
will be set with respect to him, and will not rise till he is again in the station R, that is, till the earth 
has completed one revolution. Thus whilst the earth has turned once round upon its axis from west to 
east, all the heavenly bodies in the concave sphere of the heavens will appear to have turned round 
from east to west. 

PROP. VI. The alternate succession of day and night is the effect of the revolu¬ 
tion of the earth round its axis. 

For, all the heavenly bodies appearing (Prop. V.) to move from east to west, while the earth re¬ 
volves from west to east, the sun will appear, in each revolution, to rise above the horizon in the east, 
and after describing a portion of a circle, to set in the west, and will continue below the horizon, till, 
by the revolution of the earth, it again appears in the east; and thus day and night will be alternately 
produced. 

Schol. The time of noon is found, by observing the instant when the centre of the sun is cut by the 
perpendicular hair in a meridian telescope, as described (Def. XII. Schol. 2.), or by a sun-dial. 


SECTION II. 


Of the Annual Motion of the Earth round the Sun. 

PROP. VII. The earth revolving round the sun in 365 days, 6 hours, 9 minutes, 
12 seconds, the sun appears to revolve round the earth in the same time. 

Plate 9 . Let S represent the sun, BAG the orbit of the earth, and FGHE the starry concave. Whilst the 

Fig- 8. earth is moving from A through B to C, it is manifest that, to a spectator on the earth, the sun must 
appear to move from E through F to G, in the great circle of the heavens formed by the plane of the 
earth’s orbit. In like manner, while the earth is passing from C to A, the sun will appear to pass from 
G to E. 

Schol. 1 . It is manifest that the circle in which the sun appears to move is the same, in which the 
earth w ould appear to move to a spectator in the sun. Hence the apparent place of the sun being 
found, the true place of the earth in its orbit is known. 

Schol. 2. The orbit in which the earth revolves round the sun is not a circle but an ellipse, having 
the sun in one of its foci. For the computations of the sun’s place, upon this supposition, allowing for 
the disturbing forces of the planets, are found to agree with observations. See Prop. XXXIII. 

Def. XXI. The circle which the sun appears to describe annually in the concave 
snliere of the heavens, is called the Ecliptic. 

Def. XXII. A portion of the heavens, about 16 degrees in breadth, through the 
middle of which passes the ecliptic, is called the Zodiac. 

Schol. Within this zone lie the orbits of all the planets. 


Chap. II. 


OF THE EARTH. 


161 


Def. XXIII. The stars in the Zodiac are divided into 12 Signs ; Aries, Taurus, 
Gemini, Cancer, Leo, Virgo, Libra, Scorpio, Sagittarius, Capricorn, Aquarius, Pisces. pi ( ? 
Figures, representing these signs, are drawn upon the celestial globe, in that portion Fig %. * 
of its sphericaRsurface, which corresponds to the portion of the concave sphere of 
the heavens, in which the stars belonging to each sign are respectively placed. 

PROP. VIII. The axis of the earth in every part of the earth’s revolution about 
the sun, makes, with the plane of its orbit, that is, of the ecliptic, an angle of 66^ 
degrees. 

Let BA represent the plane of the ecliptic or earth’s orbit,seen edgewise; S the sun; and P p pro- Plate y 
duced the axis of the equator. If the earth be at S, its axis is not perpendicular to the plane of the 
ecliptic, but makes an angle %vith it, PSA, of about 66° 30'. In any other part of its orbit, as M, or X, the 
axis of the earth is still inclined to the plane of the ecliptic in the same angle. 

Cor. 1. The axis, in any one part of the orbit, is in a position parallel to that in which it was at Plate 9, 
any other part of the orbit. Supposing the line FG to represent the situation of the axis of the earth Fi §. 6 
when at DFG, and to be parallel to the line HI; then when the earth is at dfg , or any other part of its 
orbit, its axis fg will still be parallel to the same line HI ; therefore f g is parallel to FG. 

Cor. 2. The planes of the equator and ecliptic make with each other an angle of 23^ degrees 
nearly. 

Schol. The obliquity of the ecliptic is not permanent, but is continually diminishing, by the eclip¬ 
tic approaching nearer to a parallelism with the equator, at the rate of about 1 a second in a year, or 
from 50" to 55" in 100 years. The inclination, in 1820, was 23° 27' 57", nearly. The diminution of 
the obliquity of the ecliptic to the equator is owing to the action of the planets upon the earth, espe¬ 
cially the planets Venus and Jupiter. The whole variation, it is said, can never exceed 2° 42', when 
it will again increase. 

The obliquity of the ecliptic may be thus found. Observe with a good instrument, very accurately 
divided, the meridian altitude of the sun’s centre, on the days of the summer and winter solstice, then 
the difference of the two will be the distance between the tropics; the half of which will be the obli¬ 
quity sought. 

By the same method, the declination of the sun for every day in the year may be found, and a table 
constructed. See Prop. XV1I1. 

Def. XXIV. The ecliptic being divided into twelve equal parts, each of these parts 
is called a Sign; and the names of the signs in the ecliptic are the same with those in 
the zodiac, but do not exactly correspond with them. 

Def. XXV. The two points in which the ecliptic cuts the equator are called the 
Equinoctial Points; the vernal equinox is at the first degree of Aries in the ecliptic; 
the autumnal, at the first of Libra. 

Schol. The moment of time in which the sun enters the equator may be found by observation, the 
latitude of the place of the observer being known. For in the equinoctial day, or near it, with an in¬ 
strument exactly divided into degrees, minutes, and parts of minutes, take the meridian altitude of the 
sun ; if it be equal to the altitude of the equator, or to the complement of the latitude, the sun is then 
in the equator; but if it is not equal, mark the difference, which will be the declination of the sun. 

The next day, again observe the meridian altitude of the sun, and gather from thence his declination. 

If these two declinations be of different kinds, as the one south and the other north, the equinox happens 
some time between the two observations; if they be both of the same sort, the sun has either not en¬ 
tered the equinoctial, or has passed it. And from these two observations of the sun’s declination, the 
moment of the equinox is thus investigated. 

Let CAB be a portion of the ecliptic, FAQ, an arc of the equator, and let their intersection be in A. Plate 9. 
Let CE be the declination of the sun at the time of the first observation, OD his declination in the Fl S- 9 ‘ 
second observation; the arc CO will be the motion of the sun in the ecliptic for one day. In the 
spherical triangle AEC, right-angled at E, we have the angle A which the equator and the ecliptic 
make, as also CE, the declination of the sun, known by observation, by which may be found the arc 
CA. And in the same manner in the triangle AOD the side AO is found; and thence the arc CO, which 
is the sum or difference of the arcs CA, AO. Therefore as CO is to CA, so is 24 hours to the time be¬ 
tween the first observation, and the moment of the ingress of the sun to the equinox. 

21 


OF ASTRONOMY. 


Book VII. Part I. 


163 


Def. XXVI. The points of the ecliptic which are at the greatest distance from 
the equator, are called the Solstices ; and the circles which pass through these points 
parallel to the equator, are called the Tropics ; the summer solstice is at the first of 
Cancer, the winter solstice at the first of Capricorn; the northern tropic is called the 
tropic of Cancer, the southern, of Capricorn. 

Cor. The sun is once in the year at each of the tropics, and twice at the equator. 

Def. XXVII. Circles which pass through the poles at right angles to the equator, 
or any other great circles, are called Secondaries to that circle; the secondary which 
passes through the equinoctial points, is called the Equinootial Colure . 

Def. XXVIII. That pole which is nearest the tropic of Cancer, is called the 
North Pole; that which is nearest the tropic of Capricorn, is called the South Pole. 

Def. XXIX. An imaginary line passing through the centre of the ecliptic, and 
perpendicular to the plane of it, is the Axis of the Ecliptic ; its extremities are the 
Poles of the Ecliptic , and all circles, passing through these poles, and perpendicular 
to the ecliptic, are its secondaries. 

Cor. The axis of the ecliptic makes an angle of 23^ degrees nearly with that of the equator. Com¬ 
pare Prop. VIII. Cor. 2. and Schol. 

Dkf. XXX. The Polar Circles are described by the revolution of the poles of the 
ecliptic about the poles of the equator: that which is next to the north pole, is called 
the Arctic circle ; the opposite, the Antarctic circle. 

Def. XXXI. The Declination of any heavenly body is its distance from the equa¬ 
tor ; this is either northern or southern. The degrees of declination of any body are 
reckoned upon a secondary of the equator passing through that body. 

Def. XXXII. The Right Ascension of any heavenly body is its distance from the 
first of Aries reckoned upon the equator; this is measured, by observing the arc which 
is intercepted between Aries and a secondary to the equator passing through the sun 
or star. 

Def. XXXIII. The Latitude of any heavenly body is its distance from the ecliptic, 
and the degrees of latitude are reckoned on a secondary of the ecliptic, passing 
through the body. 

Def. XXXIV. The Longitude of any heavenly body is its distance from the first 
of Aries; and is measured on the ecliptic by the arc intercepted between the first of 
Aries and the secondary of the ecliptic, which passes through the body; the longitude 
increases, as the body recedes from Aries, through the whole revolution, till it reaches 
360°, or comes again to Aries. 

Def. XXXV. Two bodies are said to be in Conjunction with each other, when they 
have the same longitude, or are in the same secondary of the ecliptic on the same side 
of the heavens, though their latitude be different; they are said to be in Opposition , 
when their longitudes differ half a circle, or they are on opposite sides of the heavens. 

PROP. IX. The axis of the heavens is perpendicular to the planes of all the circles 
which the heavenly bodies describe in their apparent diurnal motions. 

For the heavenly bodies, from the revolutions of the earth round its axis, appear to move from east 
to west in circles perpendicular to the axis. 

Cor. 1. The planes of all these circles are parallel to the equator. 

Cor. 2. The axis passes through the centres of the circles. 


Chap. II. 


OF THE EARTH. 


163 


Def. XXXVI. The celestial sphere is called right, oblique, or parallel, as the celes¬ 
tial equator is at right angles, oblique, or parallel to the horizon. 

PROP. X. In all places on the equator, the poles lie in the horizon, and all the cir¬ 
cles of daily motion make right angles with the horizon. 

For these places (by Def. XVIII.) having no latitude, the poles (by Prop. III.) are neither elevated 
above nor depressed below the horizon ; and since the equator is 90 degrees from the poles, it is at 
right angles to the horizon, and also all circles parallel to it. 

PROP. XI. Those who live at the equator are in a right sphere ; and, consequent¬ 
ly, their days and nights are always equal. 

The great circle of the celestial equator and its parallels (by last Prop.) make right angles with the () - 
horizons of all places in the earth’s equator; therefore (by Def. XXXVI.) the inhabitants! of those 1 10 ' 
places live in a right sphere. Hence, because the celestial axis PTS is in the plane of their horizon, 
and this axis is at right angles to the plane of the equator, and (by Prop. X.) passes through its 
centre and through that of all circles parallel to the equator, the plane of the horizon also passes 
through the centres of these circles ; and consequently divides the equator and its parallels into two 
equal parts. One half of these circles will therefore always be above the horizon, and the other half 
below it. But each of the heavenly bodies in its daily motion describes some one of those circles, 
and the diurnal motion of the earth is uniform; therefore any heavenly body will, in this situation, be 
just as long above the horizon as below it. And because this will be the case with respect to the sun, 
as well as any other body, in whatever part of the heavens he is seen, the days and nights at the equa¬ 
tor will always be of equal length. 

PROP. XII. At the poles of the earth, one celestial pole is in the zenith, and the 
other in the nadir ; the equator coincides with the horizon, and all the circles of daily 
motion are parallel to the horizon. 

For the latitude of the poles is 90 degrees from the equator, and the circles of daily motion are 
parallel to the equator. 

PROP. XIII. Those who live at either pole are in a parallel sphere ; they see the 
heavenly bodies carried round them in circles parallel to the horizon, and their day 
and their night continues each half a year. 

An inhabitant at P has the equator EQ, in the horizon, and all its parallel circles also parallel to the Plate 9. 
horizon. Therefore each of the heavenly bodies, in its apparent daily motion, being in some one of Fig. 10. 
these circles, must describe a path parallel to the horizon ; so that those which are above the horizon 
will never set by this motion, and those which are below it will never rise. The sun, therefore, in this 
situation, will not rise or set by the diurnal motion of the earth. But from the annual motion of the 
earth, the sun daily changes its apparent place in the heavens till it has described the circle of the 
ecliptic CL ; one half of which is above the horizon, and the other half below it, because these circles 
have a common centre T, the centre of the earth. Therefore, for one half of the year the sun will 
be in some part of CT, (hat half of the ecliptic which is above the horizon, and will daily revolve in 
circles above the horizon ; and for the other half, it will be in some part of TL, and will perform its 
daily revolutions in circles below the horizon. 

PROP. XIV. In any place between the poles and the equator, one celestial pole 
will be elevated, and the other depressed, at an angle less than a right angle ; and 
the celestial equator will make an angle less than a right angle with the horizon. 

For, since the place is not in the equator, it has some latitude ; and since it is not at either of 
the poles, its latitude is less than 90 degrees; whence (by Prop. III.) the poles are elhvated, or de¬ 
pressed, in an angle less than a right angle; and consequently the equator, which is perpendicular 
to the axis, makes an angle of less than 90 degrees with the horizon. 

PROP. XV. Those who live on any part of the surface of the earth between the 
equator and either pole, are in an oblique sphere, and have all the circles of daily 
motion oblique to their horizon. 


J64 


OF ASTRONOMY. 


Book VII. Part I. 


Plate 9. 
Fig. 10. 


Plate 9. 
Fig. 10. 

•i 


Plate 9. 
Fig. 7. 


Let HO be the horizon of a place which lies between the earth’s equator and either of its poles; 
the celestial equator EQ, and all its parallel circles, will be oblique to the horizon; and therefore 
each of the heavenly bodies, being - in some one of these circles, will appear to move in a path oblique 
to the horizon. 

PROP. XVI. When the sun, in his annual apparent course, is in the points in 
which the ecliptic cuts the equator, the day and night will be of the same length at all 
places on the surface of the earth ; but when the sun is in any other part of the eclip¬ 
tic, the days will be longer as the sun’s declination toward the elevated pole increases, 
and shorter as its declination toward the depressed pole increases. 

The plane of the horizon HO, of any place, passing through T, the centre of the sphere and also 
through the centre of the equator, divides the equator EQ into two equal parts, one half above, and the 
other half below the horizon. When therefore the sun has no declination, or is in the equator, it will 
appear in its daily revolution to describe the equator EQ,, and, therefore, during one half of the 
revolution, it will be above the horizon, and, during the other half, below it. 

But suppose the sun to have its declination toward P, the elevated pole, equal to Em; its diurnal 
apparent revolution will be in the circle m m, the centre of which is in a part of the axis above the 
horizon ; whence the plane of the horizon does not pass through the centre, and consequently the 
circle m m is divided into two unequal parts, the greater above the horizon, and the less below it. 
Therefore the sun, describing the circle m m, with an uniform velocity, in its apparent diurnal revo¬ 
lution will be longer in describing the part above the horizon, than the part below it. And this 
difference manifestly increases, as the circle of the sun’s apparent diurnal motion recedes from the 
equator, that is, as the sun’s declination toward P increases. In like manner, it may be shown that, 
the days will be shorter, as the sun’s declination toward the depressed pole increases. 

Or thus. Let AB represent the plane of the ecliptic seen edgewise ; S the sun in the focus of 
the-orbit; MO, KL, XY, the earth in different parts of its orbit. If FI, the axis of the ecliptic 
BA, were also the axis of the earth, that is, if the planes of the equator and ecliptic were coincident, 
it is manifest that the sun, the apparent annual motion of which is in the plane of the ecliptic, would 
at all times of the year appear to move in the circle of the equator, and to be equally distant from 
the poles, and consequently could produce, by its apparent motion, no varieties in the length of the 
days and nights. But the earth’s axis being inclined to the plane of its orbit, as P p, when the earth 
is at MO, the pole P will be toward the sun, and the pole p turned from it, and the reverse when the 
earth is arrived at XY. When the earth is in the middle station between B and A, in either part of 
its orbit, both the poles will be in the circle illuminated as at KL. 

In the position MO, since the sun must always illuminate one half of the globe, the light will pass 
beyond the pole P as far as F, and will extend toward the pole p no farther than I. Consequently, 
in the diurnal revolution of the earth round its axis, w'hile the earth remains in this position, all the 
parts of the globe between F andG will be illuminated, and all the parts between 1 and H will be dark. 
Farther, in this position greater portions of those parallels, which lie between the equator and the 
circle FG, will at any instant be in the illuminated, than in the dark, hemisphere; and, on the con¬ 
trary, greater portions of those w r hich lie between the circle HI and the equator, will at any instant 
be in the dark, than in the enlightened hemisphere. Consequently, any given place on the side of 
the equator toward P, will, in one diurnal revolution, be longer in the light than in the dark, and the 
reverse on the side toward p. The difference between the length of daylight and night will 
decrease on either side of the equator, as we approach toward it; and at the equator, the illumina¬ 
ted and dark portions of the circle being always equal, the days and nights will be of equal length. 
The contrary to all this will take place in the situation XY. Continual variations will take place, 
while the earth passes from MO to KL, and from KL to XY. But in the situation KL, the illumina¬ 
tion extending exactly to both poles, all the parallel circles are half illuminated, and half dark; con¬ 
sequently, "any place upon the globe will, in a diurnal revolution, have equal portions of light and 
darkness; that is, day and night will be every where of equal length. This must happen twice in 
every annual revolution. 

Cor. 1 . All bodies, which are on the same side of the equator with the spectator, continue longer 
above the horizon than below it, and vice versa. 

Cor. 2. As the orbits of the moon and planets are inclined to the equator, a variation of the times 
of their continuance above and below the horizon will take place. 

Schol. 1 . When the sun is very near either of the tropics, the days do not appear of different 
lengths, for the circles of apparent diurnal motion are so near to each other, that they cannot be sen¬ 
sibly distinguished. 


Chap. II. 


OF THE EARTH. 


165 


Schol. 2. The different degrees of heat at different seasons of the year are owing partly to the 
different lengths of the days, and partly to the different degrees of obliquity with which the rays fall 
upon the atmosphere at different altitudes of the sun. 

PROP. XVII. When the sun, or any other heavenly body, is in the equator, it 
rises in the east, and sets in the west. 

For it then rises and sets in the points in which the equator cuts the horizon; that is, because 
the equator is at right angles to the meridian, which passes through the north and south points, in 
the points of east and west. 

Cor. In north latitude, those bodies, which have north declination, rise between the east and 
north ; those, which have south declination, rise between the east and south. 

PROP. XVIII. When the declination of the sun is toward the elevated pole, its 
meridian altitude is equal to its declination added to the elevation of the celestial 
equator; when its declination is toward the depressed pole, its meridian altitude is 
equal to its declination, substracted from the elevation of the equator. p]ate y 

Let HO be the horizon, T the earth, P and S the celestial poles, Z the zenith, 1ST the nadir, EQ, the Fig- 10 - 
equator. If the sun be at C, having its declination toward the elevated pole P, when it arrives at the 
meridian PS, its meridian altitude CH is equal to the sum of CE, its declination, and EH, the elevation 
of the equator. If the sun be at I, having its declination toward the depressed pole S ; when it arrives 
at the meridian, its altitude IH is equal to the difference of EH, the elevation of the equator, and El, 
the sun’s declination, as appears from the figure. 

PROP. XIX. When the declination of a heavenly body toward the elevated pole is 
equal to the latitude of any place, the body will pass through the zenith of that place ; 
and when its declination toward the depressed pole is equal to the latitude, it will pass 
through the nadir. 

Any star or planet which passes through Z, the zenith, in its apparent diurnal revolution, must de¬ 
scribe the circle Zz; whence its distance from the equator or declination will be EZ. But EZ is the 
distance of the zenith from the equator, which, because the elevation of the equator is equal to the 
complement of latitude, (Prop. IV.) is equal to the latitude. In like manner the reverse may be 
proved. 

PROP. XX. A heavenly body, seen from any place, will never set from the diurnal 
motion of the earth, if the complement of its declination toward the elevated pole be 
equal to, or less than, the latitude of the place ; and it will never rise, if the comple¬ 
ment of its declination toward the depressed pole be equal to, or less than, the 
latitude. 

Let PD, which is the complement of declination of a body at D, and also the distance of the body at Plate 9. 
D from the pole, be equal to PO, the elevation of the pole, or (by Prop. III.) the latitude; it is manifest Fig. 10 - 
that the body at its lowest depression will be no farther from the pole than the horizon is; that is, will 
never be below it. In like manner the reverse may be shown. A parallel of declination, as DO, at a 
distance from the elevated pole equal to the latitude of the place, is called the circle of perpetual appari¬ 
tion ; and a parallel, as H ft, at the same distance from the other pole, the circle of perpetual occultation. 

Schol. The latitude of a place may be found, by observing the greatest and least altitude of a fixed 
star that never sets. 

Let A be a star near the north pole, which in its daily motion describes the circle AB without setting. pj ate 9 
A quadrant being placed in the plane of the meridian, or along the meridian line, observe its altitude Fig. 10 
when it is at A, and afterwards when at B ; the difference of these altitudes is AB. Aad since the star, 
in its revolution about the pole, is always at equal distances from it, if AB be bisected in P, this point 
will be the pole, and consequently PO will be the elevation of the pole. But since the lengths of the 
arcs AO, BO, have been found by observation, their difference AB, and the half of this difference, AP, 
or BP, is known; and PO is equafto BP -f BO, or to AO — PA. Whence the elevation of the pole, 
that is, the latitude, is equal to the sum of the least altitude added to half the difference of the greatest 
and least altitude, or it is equal to the remainder arising from subtracting half the difference of the 
greatest and least altitudes from the greatest altitude. 

Or, the latitude may be found from the sun’s meridian altitude and declination. If the sun’s meridian 
altitude, found by a quadrant, be CH, this altitude is equal to the sun’s declination CE, added to the 




166 


OF ASTRONOMY. 


Book VII. Part I. 


elevation of the equator EH. Therefore, if CE, the declination toward the elevated pole, be taken 
from the meridian altitude, the remainder EH will be the elevation of the equator. But since the ele¬ 
vation of the equator is the complement of latitude, the latitude is the complement of the elevation of the 
equator. This elevation, therefore, being found, the latitude of the place is known. 

Or, the latitude of a place is equal to the sun’s meridian zenith distance, added to his declination, 
ivhen he passes the meridian between the zenith and the equator. 

Exr. 1. To find the latitude from an observation of the sun's altitude, Aug, 7, 1776, at the Observa¬ 
tory at Cambridge [in England]. 

Apparent meridian altitude of the sun’s lower limb - - 53° 46' 8'' 

Sun’s apparent semi-diameter, from the Ephemeris - - 0 15 50 


Apparent altitude of the sun’s centre - 

- 54 

1 

58 

Deduct for refraction.- 

- 0 

0 

41 

Altitude of the sun’s centre. 

54 

1 

17 

Zenith distance of the sun’s centre is found by subtracting ) 
the last altitude from 90° $ 

35 

53 

43 

Add the sun’s declination - ... 

16 

13 

57 


Latitude of the place - - - - - 52 12 40 

Exp. 2. Dec. 1, 1793. The observed meridian altitude of Sirius was 59° 50'; required the latitude. 

Observed latitude - ----- 59° 50' S. 

Therefore, Zenith distance - - - - - - 30 10 N. 

Declination of Sirius - - - - - - 16 27 S. 

Consequently, the latitude required - - - - - - 13 13 N. 

Df.f. XNXVII. The two tropics and two polar circles upon the surface of the earth, 
divide it into five parts, called Zones ; the torrid zone lies between the two tropics ; 
the temperate zones between the tropics and polar circles $ and the frigid zone 9 be¬ 
tween the polar circles and the poles. 

PROP. XXI- At any place in the torrid zone the sun is vertical twice every year. 

The sun in passing from the equator to the tropic of Cancer, 234 degrees from the equator, has 
every northern declination from 0 to 23§ ; and every place between the equator and the tropic of 
Cancer has some northern latitude between 0 and 23^ ; therefore, in some part of its course from the 
equator to the tropics of Cancer, the sun must have a declination equal to the latitude of every place 
between the equator and the tropic ; whence it must be once in the zenith of every such place in its 
course toward the tropic of Cancer. For the same reason it must be once in the zenith of every such 
place in its course from the tropic to the equator. The like may be shown on the southern side of 
the equator. 

PROP. XXII. The sun is vertical once every year at the places which lie in the 
tropics. 

For the sun’s declination is then 23§ degrees, equal to the latitude of the tropics. 

PROP. XXIII. At the polar circles, the longest day and the longest night is 24 
hours. 

When the sun is in the tropic of Cancer, the complement of its declination toward the elevated pole 
is 664 degrees, equal to the latitude of the arctic polar circle; on this day, therefore, (by Prop. XX.) 
the sun will not set. When the sun is in the tropic of Capricorn, the complement of its declination to¬ 
ward the depressed pole will be 664 degrees, equal to the latitude of the arctic pole ; whence the sun 
will not rise during that day. The same may be shown with respect to the antarctic circle. 

PROP. XXIV. The longest day, and the longest night, are each of them more than 
24 hours within the frigid zone. 

For, while the sun’s complement of declination toward the elevated pole is less than the latitude of 
the place, the sun will not set; while the complement of declination toward the depressed pole is less 






Chap. II. 


OF THE EARTH. 


167 


than the latitude of the place, it will not rise ; but this must be the case with respect to every place 
within the frigid zones, in some part of the sun’s course toward the tropics. 

PROP. XXV. The sun is never vertical to any place in either of the temperate 
zones. 

For the latitude of all places in the temperate zone is greater than any declination of the sun. 

PROP. XXVI. The longest day, and the longest night, in any part of the temper¬ 
ate zones, are less than 24 hours; and the days and nights will be longer, the nearer 
the place is to the polar circles. 

For the complement of the sun’s declination can never be less than, or equal to, the latitude of any 
place in the temperate zones; whence the sun will rise and set every day within these zones. But 
the farther any place is removed from the equator, the nearer the latitude approaches to an equality 
with the complement of the sun’s greatest declination, when the day is 24 hours; that is, at the polar 
circles. 

PROP. XXVII. At different places, the hour of the day differs in proportion to 
the difference of longitude ; 15 degrees of longitude making the difference of one 
hour in time, 15" one minute of time, 15" one second of time; and it is seen at any 
given place sooner than at places which lie to the west of it, and later than at places 
which lie to the east of it. 

The sun in its daily apparent motion, which is from east to w r est, must arrive at the meridian of any 
given place, as London, sooner than it will arrive at the meridian of any place which lies to the w r est 
of London, and later than at the meridian of any place to the east of London; that is, since it is noon at 
any place when the sun is in its meridian, it will be noon at London sooner than at places west, and 
later than at places east of it. 

For example, if any place lies 15 degrees east of London, that is, has 15 degrees of eastern longi¬ 
tude from London taken as the first meridian, the sun will be one hour sooner at its meridian than at the 
meridian of London ; for, since the sun every day appears to make a complete revolution from any me¬ 
ridian to the same, in 24 hours, it will in every hour describes a 24th part of the circle, that is, 15°. 
And since a minute of a circle is a 60th part of a degree, and a second of a circle a 60th part of a minute, 
and 15' the 60th part of 15°, and 16" the 60th part of 15', the sun will move at the rate of 15' in every 
60th part of an hour, and 15" in every 60th part of a minute, that is,, in every minute or second of time. 
Consequently, it will be noon one minute or one second sooner at a place which is 15' or 15" east of 
London, than at London, 

PROP. XXVIII. The difference of longitude at two places may be found by ob¬ 
serving, at the same time from both places, some instantaneous appearance in the 
heavens. 

If the eclipse of Jupiter’s innermost satellite, on the instant of its immersion into the shadow of 
Jupiter, be observed by two persons at different places, it will be seen by both at the same instant. But 
if this instant be half an hour, for example, sooner at one place than at the other, because the places 
differ half an hour in their reckoning of time, their difference of longitude (by Prop. XXVII.) is 7° 30'. 

Schol. From tables of eclipses correc tly calculated for any place, the longitude of any place may be 
found by one observer. But such observations can only be made with certainty by land, on account of 
the motion of a ship at sea. In order to determine accurately the longitude at sea, it is necessary to 
have a clock which shall not be sensibly affected by difference of climate, difference of gravity at dif¬ 
ferent places, or the motion of the ship. Such a clock, set for the meridian of London, would constant¬ 
ly show the hour of the day at London, which it is easy to compare with the hour of the day where the 
ship is, found by observations of the sun or stars. 

PROP. XXIX. Those who live in opposite semicircles of the same meridian, 
hut in the same circle of latitude, have opposite hours of the day, but the same 
seasons. 

Being both on the same side of the equator and at the same distance from it, when the sun's deciina- 


168 


OF ASTRONOMY. 


Book VII. Part I. 


tion makes it summer or winter in one of the places, it will be the same at the other; but because they 
are distant from each 180 degrees of longitude, when it is noon at one place it will be midnight at the 
other; these are called Periosci. 

PROP. XXX. Those who live in opposite circles of latitude, but in the same 
semicircle of the meridian, have opposite seasons of the year, but the same hour of 
the day. 

When the sun has declination toward the north pole, it will be summer to those who live in the 
northern circle of latitude, and winter to those who live in the southern circle of latitude. But, 
having the same longitude, their hours of the day will be the same; these are called Antceci. 

PROP. XXXI. Those who live in opposite circles of latitude and opposite semi¬ 
circles of the meridian, have both opposite seasons of the year, and opposite hours 
of the day. 

Because they are in opposite latitudes, they will have opposite seasons ; and because they are in 
opposite semicircles of the meridian, they will have noon when it is midnight at the other; these are 
called Antipodes. 

Def. XXXVIII. Twelve secondaries to the celestial equator being conceived to 
be drawn at equal distances from each other, that is, dividing the equator into 24 equal 
parts, and the meridian of any place being made one of these secondaries, they are 
called Hour-Circles of that place. Compare Prop. XXVII. 

PROP. XXXII. If the celestial sphere had an opaque axis, the shadow of the 
axis would always be opposite to the sun; and when the sun was on one side of any 
hour-circle, the shadow of the axis would fall upon the opposite side of the same 
hour-circle. 

For all the hour-circles, being secondaries to the equator, pass through the poles, and the celes¬ 
tial axis is in the plane of every hour-circle. And the shadow of any opaque body, being opposite 
to the sun, is in the plane with the sun. Therefore in whatever hour-circle the sun is, the shadow 
of the supposed opaque axis would be in the plane of that circle and opposite to the sun; that is, while 
the sun is in one semicircle of any hour-circle, the shadow of the axis would fall upon the opposite 
semicircle. • 

Cor. Hence as the sun performs its apparent course from east to west, the shadow of the suppos¬ 
ed axis would move from west to east. 

Schol. The gnomon of a sun-dial represents the supposed axis, and hence its shadow is a measure 
of time. 

To construct a Horizontal Dial. 

In every sun-dial the gnomon, when fixed, is parallel to the earth’s axis. Now when the sun is 
in the meridian of any place, the 12 o’clock hour-circle is perpendicular to the plane of the horizon, 
and the arc from the pole to this plane is equal to the latitude of the place ; and the one o’clock hour- 
circle makes an angle at the pole with it, of 15°, and forms the hypothenuse of a right-angled triangle 
to the above perpendicular, and the base is the arc measuring the angle between 12, and 1 o’clock; 
therefore wedbave, by Spherical Trigonometry, Rad: Sin. L:: tan. 15° : tan. of the hour-angle between 
12 and 1 o’clock. If instead of 15, we substitute 30, 45, &c. we get the angles between 12 and 2, 3, 
&c. o'clock ; the same may be done for the half hours or other divisions. 

Note. The rational and sensible horizons are, in this case, supposed coincident, which, on account 
of the sun’s great distance, will not occasion any sensible error. 

To construct a Vertical South-Dial. 

In doing this we must conceive a plane passing through the centre of the earth perpendicular 
both to the horizon and meridian; and on the south side, lines must be drawn from the centre to 
the points where the hour-circles cut that plane. In finding these points, we say, as Rad : co. s. Lat : : 
tan. 15°: tan. of the hour-angle between 12 and 1 o’clock. For the arc of the meridian, from the pole 
to the plane, is equal to the complement of latitude. The other hour-angles &c. must be obtained 
in the same way as in the last. 

On the subject of Dialing, see Ferguson’s Lectures. 


* 


Chap. II. 


OF THE EARTH. 


169 


PROP. XXXIII. The orbit in which the earth revolves about the sun is elliptical. 

It is known from observation, that the apparent motion of the sun, that is, the real motion of the 
earth, in the ecliptic is not uniform. But by the universal law of bodies revolving 1 about a centre, if 
its orbit were circular, its velocity must be uniform; since ('Book II. Prop. LXXII.) it must describe Plate 9, 
equal areas in equal times. Whereas, if its orbit be an ellipse, and the sun be placed in one of the Fl S- 8i 
foci , the same law will require (see Book II. Prop. LXVIII.) that its velocity should not be uniform, 
but that in passing through its greatest distance C, to its least distance A, it should be accelerated, 
and in passing from the least distance A to the greatest C, it should be retarded. Since then the mo¬ 
tion of the earth is in fact thus retarded and accelerated in different parts of its orbit, it is manifest, 
thaj. its orbit is eliptical. 

Def. XXXIX. The greatest distance of the earth or any other planet from the 
sun, is called its Aphelion ; its nearest distance, its Perihelion ; the longer axis of 
the ellipse is called the Linea Apsidum, the aphelion is also called the Summa Apsis, 
and the perihelion the Ima Apsis. 

Def. XL. The Eccentricity of the orbit of the earth, or any planet, is the distance 
between the sun and the centre of the elliptical orbit. 

PROP. XXXIV. The sun is eight days longer in performing its apparent course 
through the six northern signs, than through the six southern signs. 

Let ABCD be the orbit of the eartb, S the sun, and EFGII the ecliptic. While the earth moves Plate 1* 
in its orbit from B through C to D, the sun appears to move in the ecliptic from F through G to II, * '»• 6 
passing through the six northern signs ; and while the earth passes from D through A to B, the sun 
appeal's to move from H to F, through the six southern signs. Now the line I IF bisects the circle 
EFGH, but divides the ellipse ABCD unequally. And, while the sun appears to pass through the 
northern signs, the earth passes through more than half its orbit; and while the sun appears to pass 
through the southern signs, the earth passes through less than half its orbit. Therefore, if the velocity of 
the motion of the earth were uniform, the sun must appear to be longerrin pa e sing through the six north¬ 
ern than the six southern signs. But whilst the earth is passing though the greater part of its orbit 
BCD, it is farther from the sun, and consequently moves slower than in the lesser part DAB. On both 
these accounts, the sun’s apparent motion is slower in the northern signs than the. southern; the dif¬ 
ference is found by observation to be about eight days. 

PROP. XXXV. The apparent diameter of the sun is greater in winter than 
summer. 

It is found by observation, that the diameter of the sun in winter is 32 minutes, 35^ seconds ; in summer, 

31' 31". And his mean apparent diameter is 32' 3'' according to Sir I. Newton, in his theory of the Moon. 

Cor. Hence it appears, that the earth, at the winter solstice, or Capricorn, is in its perihelion. 

Schol. 1. The difference between summer and winter in the degrees of heat, is owing chiefly to 
the different heights to which the sun rises above the horizon, and the different lengths of the days. 

When the sun rises highest, in summer, its rays fall less obliquely, and consequently more of them 
fall on the earth's surface than in winter; and when the days are long, and the nights short, the earth 
and air are more heated in the day than they are cooled in the night, and the reverse. 

Schol 2. The doctrine of the Sphere having been explained in the preceding propositions, some of 
the more useful Problems to be performed on the Terrestrial and Celestial Globes are here subjoined. 

Problem I. To find the latitude of any place. Bring the place to the graduated side of the fixed 
brass meridian ; the degree, under which it is found, is its latitude. All places under the same de¬ 
cree are in the same latitude. Thus the latitude of London is 5H° north, that of the Cape of Good 
idope 34° south. 

Prob. II. To find the longitude of any place. Bring the place to the fixed meridian; the distance 
of this meridian from the first meridian, measured on the equator, is the longitude of The place. The 
longitude of Boston in New England is 70i° west, or 4 hours, 42 minutes in time. That of Rome 
12|° east, or 50 minutes in lime. 

Prob 111. To rectify either globe to the latitude of any place, the zenith , and the snips place. If the 
place be in the northern hemisphere, raise the north pole above the horizon; but if the place be 
in the southern hemisphere, raise the south pole. Then move the meridian up and down in the notch- 


170 


OF ASTRONOMY. Book VII. Part I. 

es, till the degree of the place’s latitude, counted upon the meridian, below the pole, cuts the hori¬ 
zon ; and then the globe is adjusted to the latitude of the place. 

Having elevated the globe according to the latitude of the place, count the same number of degrees 
upon the meridian, from the equator toward the elevated pole, and that point will be the zenith or 
vertex of the place. To this point of the meridian screw the quadrant of altitude, so that its gradu- 
ated edge may be joined to the said point; then is the globe rectified for the zenith. 

Bring the sun’s place in the ecliptic to the meridian, and set the hour-index to 12 at noon ; and 
then the globe will be rectified for the sun’s place. 

Prob. IV. To determine the difference of time in different places. Find the longitude of each place, 
and reduce the difference into time, allowing an hour for every 15 degrees, and proportionally for less¬ 
er parts; the difference of time will be found. If the place lies westward of another, it has its noon 
later than that other; if eastward, sooner. The longitude of Rome is 12^° east, that of Constantino¬ 
ple 29°, the difference is 17?,°, consequently the difference of time between Rome and Constantinople 
is 1 hour and 10 minutes. 

Prob. V. The latitude and longitude of anyplace being known, to find the place on the globe. Bring 
the degree of the equator which expresses the given longitude to the fixed meridian, then find the 
given latitude on the meridian; under this point is the place sought. 

Prob. VI. To find the distance between any two places , and their bearing , or relative situation with 
respect to the points of the compass. Rectify the globe to the latitude of one of the places, and bring the 
place to the fixed meridian. Then fix the quadrant of altitude to the uppermost point of the meridian, 
and putting its lower end between the horizon and the globe, slide it along till it passes through the other 
place. The number of degrees on the quadrant between the two places, will give their distance, allowing 
69^ English miles for each degree; and the number of degrees upon the horizon between the meridian 
and the quadrant, will give the bearing of the second place with respect to the first. Thus the bearing 
of the Lizard Point from the island of Bermudas is nearly E. ]N T . E. 

Prob. VII. To find the right ascension and declination of the sun , or any star. On the celestial globe 
find the day of the month under the ecliptic, against which is the sun’s place, or find his place by an 
ephemeris ; bring that point under the meridian, and the degree which is over the point is the sun’s 
declination, and the degree of the equator then under the meridian will be the sun’s right ascension. 
A star’s declination and right ascension are found, by bringing the star on the globe to the merid¬ 
ian, and proceeding as with respect to the sun. The sun’s declination, April 19, is 11° 14'north, and 
his right ascension 27° 30'.—The right ascension of Sirius is 99°, its declination 16° 20' south. 

Prob. VIII. To find what stars pass over, or near, the zenith of any place. Having found the lati¬ 
tude of the place on the terrestrial globe, all those stars on the celestial globe, which pass under the 
same degree of the meridian with the given latitude, become vertical at that place. 

Prob. IX. To find what stars never rise , or never set , in a given place. The globe being rectified 
for the given place, those stars which do not pass under the wooden horizon, never set; those which 
do not come above it, never rise. 

Prob. X. To represent the appearance of the heavens at any time. Rectify the globe to the latitude ; 
bring the sun’s place in the ecliptic to the meridian, and set the horary index to the upper 12th 
hour; then turn the globe till the index points to the given hour. The north pole of the globe 
must be set to the north in the heavens, then will all the stars upon the globe correspond to their 
places in the heavens, so that an eye at the centre of the globe would refer every star upon its sur¬ 
face to the place of the stars in the heavens. By comparing, therefore, the stars in the heavens with 
their places on the globe, a person will easily get acquainted with all the stars. 

Example. The situation of the stars at London on the 9th of February, at nine o’clock in the even¬ 
ing is as follows; Sirius, or the dog-star, is on the meridian, its altitude 22°; Procj'on, <ho little dog- 
star, 1G° toward the east, its altitude 43°; about 24° above this last, and a little more toward the east, 
are Castor and Pollux ; S. 65° E. and 35° in height is Regulus, or Cor Leonis; exactly in the east, 
and 22° high, is Deneb in the Lion’s tail; 30° from the east toward the north, Areturus is about 3° 
above the horizon; directly over Areturus and 31° above the horizon, is Cor Caroli; in the north¬ 
east are the stars in the extremity of the Great Bear’s fail. 

Reckoning westward, we see the constellation Orion ; the middle star of the three in his belt 
is S. 20° W. its altitude 35°; nine degrees below the belt, and a little more to the west is Rigel, the 
bright star in his heel; above his belt in a straight line drawn from Rigel, between the middle and 
most northward in his belt, and 9° above it, is the bright star in his shoulder; S. 49° W. and 451 ° above 
the horizon is Aldebaran, the southern eye of the Bull; a little to the w'est of Aldebaran, are the 
Hyades; the same altitude, and about S. 70° W. are the Pleiades ; in the W. by S. is Capella in Auriga, 
its altitude 73° ; in the north-west, and 42° high, is the constellation Cassiopeia; and almost in the 
north, near the horizon is the constellation Cygnus. 

Prob. XI. The latitude of a place being given , to find the time of the sun's rising and setting , on ami 
given day , at that place. Having rectified the globe according to the latitude, bring the sun’s place 
in the ecliptic to the graduated edge of the meridian, and set the horary index to the upper 12. Then 


Chap. II. 


OF THE EARTH. 


171 


turn the globe to bring the sun’s place to the eastern part of the horizon, the index will point to 
the hour at which the sun rises; on the western side, to the time of its setting. On the 5th of June 
the sun rises at 3h. and 40 min. and sets at 8 h. and 20 min. The length of the day is 16 h. 40 min. that 
of the night 7 h. 20 min. 

Prob. XII. To find all the places on the globe to :which the sun will be vertical on a given day. Bring 
the sun’s place to the fixed meridian, and observe the point of the sun’s declination; all the places 
which in turning the globe round pass under that point, have the sun vertical on the given day. 

When the sun’s declination is equal to the latitude of any place, then the sun will be vertical to 
the inhabitants of that place. 

Prob. XIII. To find the sun's amplitude. The globe being rectified for the latitude of the place, bring 
the sun’s place to the eastern side of the horizon; the arc of the horizon intercepted between that point 
and the eastern point, is the sun’s amplitude at rising. Thus on the 24th of May, the sun’s amplitude 
at rising is 36° east, and he sets with 36° western amplitude.—The amplitude increases with the 
latitude of the place. 

Pros. XIV. To find the sun's meridian altiiude. The globe being rectified for the latitude, zenith, 
and sun’s place, the number of degrees contained between the sun’s place and the zenith is the dis¬ 
tance of the sun from the vertex at noon ; the complement of which to 90 degrees, is the sun’s alti¬ 
tude. The meridian altitude of the sun on the 17th of May, at London, is 57° 55'. The altitude 
being given, bring the quadrant of altitude to meet the sun’s place and the intersection of the quadrant 
and horizon will show the azimuth. Thus on the 21st of August, at London, when the sun’s altitude 
is 36° in the forenoon, the azimuth is 60° from the south. 

Prob. XV. To find the place of any heavenly body upon the globe , its longitude and latitude being given. 
Place the first degree of the quadrant of altitude upon that degree of the ecliptic which expresses the 
given longitude, and the 90th degree on the pole of the ecliptic; the point of the globe which is 
under that degree of the quadrant which expresses the given latitude, is the place of the body ; for 
the quadrant represents a secondary of the ecliptic, an arc of which between the body and the eclip¬ 
tic is its latitude, and the arc of the ecliptic between the secondary and the first degree of Aries its 
longitude. 

Prob. XVI. To find the place of any heavenly body upon the globe, its right ascension and declination 
being given. Bring that point of the equator which expresses the given right ascension to the me¬ 
ridian ; the place sought is under that degree in the meridian, north or south, which expresses the given 
declination. 

Prob. XVII. To find all those places where it is noon, at any given hour of the day, at any given place. 
Bring the given place to the brass meridian, and set the index to the uppermost 12; then turn the 
globe till the index points to the given hour, and it will be noon to all the places under the meridian. 
When it is 4 h. 50 m. in the afternoon at Paris, it is noon at New Britain, St. Domingo, Terra Firma, 
Peru, Chili, and Terra del Fuego. As the diurnal motion of the earth is from west to east , it is plain 
that all places which are to the east of any meridian must necessarily pass by the sun before a meridian 
which is to the west can arrive at it. 

Prob. XVIII. The hour being given at any place, to tell what hour it is in any other part of the world. 
Bring the place where the time is given, under the meridian; set the hour index to the given time, 
and turn the globe till the other place comes under the meridian, and the horary index will point 
to the hour required. Thus, when it is nine in the morning at London, it is half past four in the 
afternoon at Canton in China; when it is three in the afternoon at London, it is 18 minutes past ten 
in the forenoon at Boston in America. 

Prob. XIX. To find the antceci, the periled, and the antipodes of any place. Bring the given place 
to the meridian, then, in the opposite hemisphere, and in the same degree of latitude with the given 
place will be the antceci. The given place remaining under the meridian, set the index to 12, and 
turn the globe till the other 12 is under the index; then the periceci will be under the same degree 
of latitude with the given place, and the antipodes of the given place will now be under the same point 
of the brazen meridian where the antceci stood before. 

Prob. XX. To find the two days on which the sun is in the zenith of any given place between the tropics. 
That parallel of declination which passes through the given place, will cut the ecliptic line upon the 
globe in two points, which denote the sun's place ; against which, on the horizon,,will be lound the 
days required. The sun is vertical at Barbadoes, April 24, and August 18. 

Prob. XXL The time and place being given , to find all those places where the sun is rising , setting, 
culminating ; and also where it is daylight , twilight , or dark-night. Find the place where the sun is 
vertical at the given hour, rectify for the latitude of that place, and bring it to the meridian. Then 
all the places that are in the a-esf semicircle of the horizon, have the sun rising; those in the east se- 


172 


OF ASTRONOMY. 


Book VII. Part I. 


micircle, have it setting ; those under the meridian above the horizon, have it culminating ; and all 
places above the horizon, have the sun so many degrees above the horizon, as the places them¬ 
selves are. Those places that are below the horizon, but within 18° of it, have twilight ; those lower 
than 18° have dark-night; and to those under the meridian it is midnight. 

Prob. XXII. To find the place of the moon , or any planet, for any given time. Take the nautical 
almanac, or White’s Ephemeris, and against the given day of the. month will be found the degree 
and minute of the sign which the moon or planet possesses at noon. The degree thus found, and 
marked in the ecliptic on the. globe, you may proceed to find the declination, right ascension, lati¬ 
tude, longitude, altitude, amplitude, azimuth, rising, setting, &c. 

Pf.ob. XXIII. To find the planets which cere above the horizon at sun-set , upon any given day and 
latitude. Find the sun’s place for the given day, bring it to the meridian, set the hour index to 12, 
and elevate the pole for the given latitude, then bring the sun’s place to the western semicircle of 
the horizon, and observe what signs are in that part of the ecliptic above the horizon, then look to 
the ephemeris for the day, and it will be seen what planets are in those signs, for such will be visible 
on the evening of that day. 

Prob. XXIV. To find whether Venus be a morning or an evening star. Rectify the globe for the 
latitude and sun’s place ; find the situation of Venus by an ephemeris, and stick there a small black 
patch ; bring the sun’s place to the edge of the eastern horizon. If Venus be i?i antecedentia, that is, for 
instance, in Taurus when the sun is in Gemini, she will be a morning star. 

But, if Venus be in consequentia , that is, for example, in Gemini w hen the sun is in Taurus, she will 
set after the sun, and be an evening star. 

Prob. XXV. To find all the places to which a lunar eclipse will be visible at any instant. Find the place 
to which the sun is vertical at the given time, and bring that place to the zenith, and the eclipse will 
be visible to all the hemisphere below* the horizon, because the moon is opposite to the sun. 

Schol. 1. Since lunar eclipses continue in general, for a considerable time together, they may be 
seen in more places than in one hemisphere of the earth ; for, by the earth’s rotation about its axis, 
during the time of the eclipse, the moon will rise to several places after its commencement. 

Schol. 2. We cannot, by a globe only, determine the places to which a solar eclipse is visible, because 
that eclipse does not happen to the whole hemisphere next the sun, nor does it happen at the same 
t-ime to those places where it is visible. Calculations are therefore necessary. 

SECTION. III. 

Of Tivilight. 

PROP. XXXVI. The atmosphere by refracting the rays of light increases the 
apparent altitude of all the heavenly bodies, except when they are in the zenith. 

Plate 16. L e t AB be a portion of the earth’s surface, a b the upper part of the atmosphere over it, HO the 
Fig ' horizon of the place A, S the sun or a star seen from A. The rays of light proceeding from S on enter¬ 

ing the atmosphere are refracted toward the perpendicular (by Pi’op. XI. B. 6.), and as the air increases 
in density in approaching the earth, the light is constantly passing from a rarer to a denser medium, and 
of course constantly refracted more and more toward a perpendicular, describing a curve, the character 
of which has not been determined. Now (by Prop. LXXX. B. 6.) the body S will appear at s in a line 
which is a tangent to this curve in the point A, and as the refraction is toward the perpendicular the 
point s or apparent place of the body S will be nearer the zenith or have a greater altitude than the body 
itself. As in all other cases of refraction the larger the angle of incidence (or the less the altitude) the 
greater is the refraction or angle of deviation. In the horizon this angle is about 33', so that the w'hole 
of the sun’s disc is apparently above the horizon when it is really below it. By this means we enjoy 
the sight of the sun, in latitude 42A about 6m. longer every day in clear w’eather than we should do 
without this refraction. This elevation diminishes very fast in the first degrees of altitude, and when 
the body is in the zenith, the rays coincide with the perpendicular and (by Prop. XIV. B. 6.) suffer no 
refraction. 

Tables of refractions have been calculated by various astronomers, as Sir I. Newton, Mr. Thomas 
Simpson, Dr. Bradley, Mr. Mayer, &c. The following specimen is taken from Dr. Bradley’s table, 
which is esteemed the most correct, and chiefly used by astronomers. For the method of calculating 
these tables, see Mr. Simpson’s Dissert, p. 46, 4to. Gregory’s Astron, Vol. i. Pr. 66; and Vince’s 
Astron. Vol. i. 4to. ch. 7. 


Chap. II. OF THE EARTH. 17 * 


MEAN ASTRONOMICAL REFRACTIONS IN ALTITUDE. 


App 

Alt. 

Refraction. 

App. 

Alt. 

Refraction. 


R efraction. 

App. 

Alt. 

Refraction. 

App. 

Alt. 

1 

Refraction. 

aK 

Re raction. 

0 

/ 

// 

0 

/ 

n 

O 

/ 

// 

O 

/ 

// 

O 

// 

O 

// 

0 

33 

0 

11 

4 

47 

23 

2 

14 

35 

1 

21 

48 

51 

78 

12 

1 

24 

29 

12 

4 

23 

24 

2 

7 

36 

1 

18 

50 

48 

80 

10 

2 

18 

35 

13 

4 

3 

25 

2 

2 

37 

1 

16 

52 

44 

82 

8 

3 

14 

36 

14 

3 

45. 

26 

1 

56 

38 

1 

13 

55 

40 

85 

5 

4 

11 

51 

15 

3 

30 

27 

1 

51 

39 

1 

10 

58 

35 

88 

2 

5 

9 

54 

16 

3 

17 

28 

1 

47 

40 

1 

8 

60 

33 

89 

1 

6 

8 

28 

17 

3 

4 

29 

1 

42 

41 

1 

5 

62 

30 

90 

0 

7 

7 

20 

18 

2 

54 

30 

1 

38 

42 

1 

3 

‘65 

26 



8 

6 

29 

19 

0 

45 

31 

1 

35 

43 

1 

1 

'68 

23 



9 

5 

48 

20 

2 

35 

32 

1 

31 

44 

0 

59 

170 

21 



10 

5 

15 

21 

2 

27 

33 

1 

28 

45 


57 

72 

18 






22 

2 

20 

34 

1 

24 



• 

75 

15 




PROP. XXXVII. The atmosphere above the horizon is enlightend by the rays of 
the sun, when the sun itself is below the horizon. 

Let ADL be the surface of the earth ; CBM, the surface of the atmosphere ; A, any place upon the pi a te y, 
earth; PABN, the sensible horizon. When the sun is at G, any point below the horizon, it cannot be Fig. It. 
directly seen by a spectator at A. But, because rays from the sun at G can pass to the part of the 
atmosphere above the sensible horizon of the place A, this part of the atmosphere will be illuminated 
before the sun rises, or after it sets, and will become visible by reflection to the spectator at A ; that is, 
twilight will be produced. 

Cor. The atmosphere is the cause of the apparent elevation of the heavenly bodies above their true 
places. 

PROP. XXXVIII. When the evening twilight ends, or the morning twilight be¬ 
gins, a ray of the sun reflected from the highest part of the atmosphere describes, after 
reflection, a line, which is the plane of the sensible horizon. 

As the sun is depressed, the extreme ray of light from the sun gradually recedes from C toward B, Plate. 9 . 
till at last it touches the horizon at B, from whence it is reflected in the direction BA, the plane of the Fi S- H. 
horizon. 

PROP. XXXIX. If the time of the beginning of the morning or end of the evening 
twilight is known, the height of the atmosphere may be determined. 

It is found, that at the beginning of the morning, or the end of the evening twilight, the sun is 18 Plate 1 <>. 
degrees below the horizon. And the sun’s centre being 18° below the horizon, when the first ray of Fig. 3. 
light appears, his upper limb must be depressed 18° — 16' = 17° 44', and if the sun were a point, it 
would be only 17° 44'below the horizon at the beginning of twilight, or, subtracting the refraction 
which takes place in passing from S to 1), 17° 1 T. Hence AED = 17° It', and its half AEB = 8 ° 35 a, 
and if the rays DB, BA, were not refracted, the upper part of the atmosphere would be at B, but the ray 
SD, tangent to the surface at D, is bent into a curve D b , and at b is reflected and moves in a similar curve 
from b to A. Draw b c tangent to the curve b A in b to cut AB in c, and A a parallel to c b to cut EB in 
a. Hence B c b = BA a — the horizontal refraction 33', since the tangent at b lies in the direction of 
the ray on its reaching the atmosphere, and it is evident that the real height of the atmosphere falls 
between HB and H a, to calculate which we have in the triangle AEB all the angles, viz. EAB = 90°, 

AEB = S°35A', ABE = 81° 24§', and AE — 1 (radius of the earth), to find EB = 1.01135, and HB = 

.01135. And in the triangle AE u we have EA a = 89° 27', E a A = 81° 57§' = AE a = 8°J55£', to find E a 
= 1.00989, and H a = 00989 Hence H b is greater than 0.00989 and less than 0.01135, the radius of 
the earth being 1 . If most of the refraction take place very near the earth at D and A, the point b 
will fall near a. On the contrary, if it take place very near the upper part of the atmosphere, the 





























174 


OF ASTRONOMY. 


Book VII. Part I. 


Plate 9. 
Fig. 2. 


Plate. 9. 
Fig. 13. 


point b will fall near B. If we suppose, as usual , that the curve A b is the arc of a circle, we shall 
have, after drawing the chord A 6, 6Ac = £tcB =16$', (as b c B is the external angle of the isosceles 
triangle A be). Hence EA b = 89° 43$', AE 6=8° 351', E£> A =81° 41', and, as EA = 1, we find the 
side E6 of the triangle EA b to be 1.0106. Hence H b = 0.0106, or about 42 statute miles. 

Schol. Dr. Keill, in his Lectures on Astronomy, observes, that it entirely owing to the atmosphere 
that the heavens appear bright in the day time. For without it, only that part of the heavens would 
be luminous in which the sun is placed ; and if we could live without air, and should turn our backs to 
the sun, the whole heavens would appear as dark as in the night. In this case also we should have no 
twilight, but a sudden transition from the brightest sunshine to dark night immediately upon the set¬ 
ting of the sun, which would be extremely inconvenient, if not fatal to the eyes of mortals. See Keill’s 
Astron. Lect. xx. 

PROP. XL. The twilight is longest in a parallel sphere, and shortest in a right 
sphere; and in an oblique sphere, the nearer the sphere approaches to a parallel, the 
longer is the twilight. 

In a parallel sphere, the twilight will continue till the sun’s declination toward the depressed pole is 
18°, but in this sphere his declination is never more than 23$ degrees ; whence the twilight will only- 
cease, whilst the sun’s declination is increasing from 18° to 23$°, and decreasing again till in its decrease 
it becomes 18°. The twilight is here caused by the annual motion of the earth. In a right sphere, 
the sun appears to be carried, by the daily motion of the earth, in circles perpendicular to the ho¬ 
rizon ; whence it is carried directly downward by the whole daily motion, and will arrive at 18° below 
the horizon, the soonest possible ; w'hereas, in an oblique sphere its path is oblique to the plane of the 
horizon, and therefore will be longer before it is descended 18 degrees below the horizon ; and the 
difference of the time of twilight will increase w ith the degree of obliquity. As the sun sets more ob¬ 
liquely at some parts of the year than others, the twilight varies in its duration. 

SECTION. IV. 

Of the Equation of Time. 

PROP. XLI. The time in which the sun completes one apparent diurnal revolu¬ 
tion, is greater than that in which the earth revolves round its axis. 

If the earth turns round its centre T in the direction RCB, and at the beginning of one revolution 
the sun was seen at Z, in the meridian ; from its apparent annual motion it will, after the diurnal rev¬ 
olution of the earth is completed, be seen advancing in its orbit toward E. The earth, therefore, must 
perform more than one revolution, and the spectator at R, after returning to the station from which 
he set out, must advance forward to e, before the sun will be again in the meridian. 

PROP. XLII. The obliquity of the ecliptic to the equator would cause the daily 
increments of the sun’s right ascension to be unequal, although the sun’s motion in the 
ecliptic were uniform. 

Let E be the first degree of Aries, EQ, an arc of 90° of the equator, EC the same of the ecliptic, 
and CQ, an arc of the solstitial colure between Cancer and the equator. At E the sun has neither lon¬ 
gitude nor right ascension ; these may therefore be considered as equal, when the sun sets out from 
Aries. At C, the longitude is equal to the right ascension ; for both EC and EQ are by supposition 90 
degrees of great circles of the same sphere. But if the sun be any where between the first of Aries 
and the first of Cancer, as at S, the longitude will be greater than the right ascension ER. For SR be¬ 
ing an arc to the secondary of the equator parsing through the sun, ES is the longitude, and ER the 
right ascension ; but ES is greater than ER, because the angle at R is a right angle, but the angle at S 
an acute angle. Now, if the sun be supposed to move uniformly in the ecliptic, or to describe equal 
arcs in equal times, the daily increments of longitude will be equal to one another; and consequently, 
since at the iwo extremes E and C the longitude and right ascension are equal, and the longitude is 
supposed to increase uniformly, if the right ascension also increased uniformly, they would at all times 
be equal. But at S, R,-or any o*her points in the same secondary, between the first of Aries and Cancer, 
the longitude is greater than the right ascension; the daily increments of right ascension arc therefore 
unequal. 


Chap. II. 


OF THE EARTH. 


175 


The longitude and right ascension are equal when the sun is at C and at E, the former being 90° the p] a t e 9 . 
latter 180° from Aries both on the ecliptic and equator. But between C and E, the longitude is less Fig. 14. 
than the right ascension; because ES,'opposite to the right angle R, is greater than ER opposite to the 
acute angle S, and consequently the point S is nearer Aries than the point R. But, the sun being sup,- 
posed to move uniformly, or to increase its longitude equally every day, if the right ascension also in¬ 
creased equally every day, since the longitude and right ascension are equal at C and E, they would be 
always equal. But at S, or any where between Cancer and Libra, the longitude is less than the right 
ascension; consequently, the daily increments of right ascension are not equal. In like manner it 
may be shown, that in the third quarter the sun’s longitude is greater than its right ascension, and in 
the fourth, less. 

PROP. XL.III. If the plane of the ecliptic coincided with that of the equator, the 
daily increments of the sun’s right ascension would nevertheless be unequal. 

Because (by Prop. XXXIV.) the apparent annual motion of the sun is not uniform, it would jn 
some days describe a longer arc than in others; that is, since its right ascension and longitude would 
in this case be the same, the daily increments of its ascension would be unequal. 

Dff. XLI. A Natural Day is the time the sun takes in passing from the me¬ 
ridian of any place, till it comes round to the same meridian again. 

PROP. XLIV. Any place upon the earth’s surface describes more than a circle 
round the earth’s axis in a natural day; and the arc which it describes more than a 
circle in any day, is the sun’s increment of right ascension for that day. 

While the earth revolves round its axis, any place -upon the earth’s surface describes a circle ; 
but (by Prop. XLIII.) while the sun completes its apparent diurnal revolution, any place on the earth’s 
surface will move through one circle and an arc of a second; therefore any such place describes 
more than a circle (Def. XLI.) in a natural day. 

And since both a meridian, and a secondary of the equator, passing through the poles, are per- Plate 9. 
pendicular to the equator (Def. VI. and X.), if the sun at S be in SR, the meridian of any given place, F *g- 13 - 
it is also a secondary of the equator passing through that place. In like manner, if the sun be at T, 
and TV be after a natural day the situation of the meridian of the given place, the sun will be in TV, 
which will be both the meridian of the place, and a secondary of the equator. Whence, RV being 
part of the equator, since ER was the sun’s right ascension when it was at S, and EV is its right as¬ 
cension when it is arrived at T, RV must be the increment of the sun’s right ascension for the natu¬ 
ral day in which it is advanced from S to T. And, because SR, TV, are both perpendicular to the 
equator, and any place, in one diurnal revolution of the earth, describes a circle parallel to the equator, 

RV taken in this circle will always be the same arc with RV in the equator, and therefore will be 
equal to the sun’s daily increment of right ascension. 

PROP. XLV. The natural days are not equal to one another. 

For any natural day is the time in which the earth performs one revolution round its axis, and 
such a portion of a second, as is equal to the sun’s increment of right ascension for that day ; but the 
sun’s daily increments of right ascension are unequal (by Prop. XL1I. and XLIII.); therefore the ad¬ 
ditional portion of the second revolution will sometimes be greater and sometimes less, and conse¬ 

quently the times in which the natural days are completed will be unequal. 

Dep. XLII. The Equation is the difference between mean time and apparent time. 

PROP. XLVI. If the sun were to move uniformly round the equator in the same 
time in which it appears to describe the ecliptic, its apparent daily motion would be 
a measure of mean time. 

For the natural days in that case being liable to no variation, either from the obliquity of the 
sun’s orbit, or the irregularity of its motion, must be equal. 

PROP. XLVII. The portion of time which passes between the arrival of the sun 
in the ecliptic to the meridian of any place, and its supposed arrival at the same me¬ 
ridian, if it were to move uniformly in the equator, is the equation. 

For (by the last Prop.) it would be noon by mean time at any place, when the sun, if it moved 


17G 


OF ASTRONOMY. 


Book. VII. Part I. 


Plate 9. 
Fig 6. 


Plate P. 
Fig. 13. 


uniformly in the equator, was arrived at the meridian of that place; and it is noon at the same place 
by apparent time, when the sun in the ecliptic arrives at the same meridian ; therefore the difference 
between these two arrivals is the difference between mean time and apparent time, or the equation. 

PROP. XLVIII. In the time which passes between the arrival of the sun in the 
ecliptic to the meridian of any place, and its supposed arrival at the same meridian, 
if it were to move uniformly in the equator, an arc of the equator passes under the 
meridian, which is equal to the difference between the right ascension of the sun, as 
it moves in the ecliptic, and the right ascension which the sun would have, if it moved 
uniformly in the equator. 

Let the sun be at S ; and let EC be the ecliptic, EQ the equator, and E the first of Aries; then 
if a secondary of the equator passes through the sun, SR, being at right angles to EQ, is an arc of 
that secondary, and (by Del'. XXXI1.) Ell is the sun’s right ascension, and the point R is the point in 
which the right ascension ends ; which being in the secondary of which SR is a part, that is, the 
secondary passing through the sun, arrives at the meridian at the same time with the sun. If the 
sun were to move uniformly in the equator, and were arrived at P, EP would be its right ascension, 
and consequently P would be the point in which its right ascension would end, which point P must 
arrive at the meridian at the same time with the sun, because the sun is supposed to be in that point. 
Therefore RP the* distance of the two points R and P, is an arc of the equator (passing under the 
meridian in the time specilied in the Proposition), which is equal to the difference between the real 
and supposed right ascensions of the sun, when he arrives at the meridian by his real motion in the 
ecliptic, and when he arrives at the same meridian by an uniform motion in the equator. 

PROP. XLIX. The right ascension of the sun, if it were to move unifomly in the 
equator, would at any time be equal to the longitude which it would have at that 
time if it were to move uniformly in the ecliptic, or to its mean longitude. 

For on this supposition, the sun, describing the equator with an uniform velocity in the same time 
in which it actually describes the ecliptic, its velocity would be the same with the mean velocity in 
the ecliptic. Consequently, the distance of the sun from the first of Aries in the equator would at 
any time be the same with its distance from the same point in the ecliptic, if it were to move uni¬ 
formly therein with its mean velocity; that is, its right ascension in the equator would always be 
equal to its mean longitude in the ecliptic. 

PROP. L. Ail arc of the equator, equal to the difference between the sun’s right 
ascension and its middle longitude, at any given time and place, converted into time, 
is the equation. 

It has been shown (Prop. XLVII1.) that in the portion of time which passes between the arrival 
of the sun in the ecliptic to the meridian of any place, and its supposed arrival at the meridian, if it 
were to move uniformly in the equator, an arc of the equator passes under the meridian, which is 
equal to the difference of the right ascension of the sun as it moves in the ecliptic, and the right as¬ 
cension which it would have if it moved uniformly in the equator. And it has been proved (Prop. 
XLVI1.) that this portion of time is the equation, and (Prop. XLIX.) that the right ascension which 
the sun, at any given time and place, would have if it moved uniformly in the equator, is equal to its 
mean longitude in the ecliptic. Therefore, in the equation, an arc of the equator passes under the 
meridian, equal to the difference of the right ascension of the sun in the ecliptic, and its mean longi¬ 
tude. Consequently, if this arc be converted info time, that is, if for 15° be fc taken an hour, for 15' 
one minute of time, for 15" one second of time, the equation of time will be found. 

PROP. LI. If the sun’s mean longitude be greater than its right ascension, mean 
time follows apparent time ; if its mean longitude be less than its right ascension, 
mean time precedes apparent. 

If the right ascension of the sun, as before supposed in the equator, EP, that is, (by Prop. XLIX.) 
its mean longitude, be greater than the sun’s real right ascension Eli, the supposed place of the sun in the 
equator P, will be to the east of the point P>, where the sun’s real right ascension ends. Therefore 
when this point R, at apparent noon, is come to the meridian, the point P will not be arrived at the 


Chap. II. 


OF THE EARTH. 


17 


meridian; and mean noon will be later than apparent noon. Therefore, when the sun% middle 
longitude is greater than its right ascension, mean time follows apparent. In like manner the reverse 
may be proved. 

Cor. Hence in the former case the equation is to be subtracted from the apparent time found by 
the dial, and in the latter , to be added to it, in order to obtain the mean time. 

Schol. It appears from the foregoing Propositions, that the difference between mean and apparent 
time, depends upon two causes; (1) the obliquity of the ecliptic with respect to the equator; and (2) 
the unequal motion of the earth in an elliptical orbit. The obliquity of the ecliptic to the equator 
would make the sun and clocks agree on four days of the year, viz. when the sun enters Aries, 
Cancer, Libra, and Capricorn. But the other cause which arises from his unequal motion in his orbit 
would make the sun and clocks agree only twice a year, that is, when he is in perigee and apogee ; conse¬ 
quently, when those two points fall in the beginnings of Cancer and Capricorn, or of Aries and Libra, they 
will concur in making the sun and clocks agree in those points. But the apogee, at present, is in the 
tenth degree of Cancer, and the perigee in the tenth degree of Capricorn ; and, therefore, the sun 
and clocks cannot be equal at the beginnings of these signs, nor indeed, at any time of the year, ex¬ 
cept when the equation resulting from one of the causes is just equal to that arising from the other, 
one being positive and the other negative, which happens about the 15th of April, the 15th of June, 
the 31st of August, and the 24th of December; at all other times the sun is too fast or too slow for 
equal time by a certain number of minutes and seconds, which at the greatest is 16' 14", and happens 
about the first of November; every other day throughout the year having a certain quantity ot this 
difference belonging to it, which, however, is not exactly the same in every year, but only every fourth 
3 'ear, for which reason it is necessary, where great accuracy is required, to have four tables ot this 
equation, viz. one for each of the four years in the period of leap years. The following concise 
table, adapted to the second j'ear after leap year, will always be found within about a minute of the 
truth, and therefore sufficiently accurate for common clocks and watches. 

TABLE FOR THE EQUATION OF TIME. 


Days. 

Months. 

Equation 

in 

Minutes. 

Days. 

Months. 

Equation 

in 

Minutes. 

Days. 

Months. 

Equation 
in J 

Minutes. 

Days. 

Months. 

Equation 

in 

Minutes. 

Jan. 1 

d + 

Apr. 1 

4 + 

Aug. 9 

5+ 

Oct. 27 

16— 

3 

5 

4 

3 

15 

4 

Nov. 15 

15 

5 

6 

7 

2 

20 

3 

20 

14 

7 

7 

11 

1 

24 

2 

24 

13 

9 

8 

15 

0 

28 

1 

27 

12 

12 

9 

* 


31 

0 

30 

11 

15 

10 

19 

1 — 

* 


Dec. 2 

10 

18 

11 

24 

2 

Sept. 3 

1 — 

5 

9 

21 

12 

30 

3 

6 

2 

7 

8 

25 

13 

May 13 

4 

9 

3 

9 

7 

31 

14 

29 

3 

12 

4 

11 

6 

Feb. 10 

15 

June 5 

2 

15 

5 

13 

5 

21 

14 

10 

1 

18 

. 6 

16 

4 

27 

13 

15 

0 

21 

7 

18 

3 

Mar. 4 

12 

* 


24 

8 

20 

2 

8 

11 

20 

1 + 

27 

9 

22 

1 

12 

10 

25 

2 

30 

10 

24 

0 

15 

9 

29 

3 

Oct. 3 

11 

* 


19 

8 

J uly 5 

4 

6 

12 

26 

1 + 

22 

7 

11 

5 

10 

13 

28 

2 

25 

6 

28 

6 

14 

14 

30 

3 

| »28 

5 



19 

15 




Those columns that are marked -f-,show that the clock or watch is faster than the sun; and those marked _-that, it ?s 

liower. See Ferguson’s Astronomy, ch. 13. Phil. Trans, vol. 54, 

23 































178 


OF ASTRONOMY. 


Book VII. Part I. 


Plate 9. 
Fig. 12. 


Plate. 9. 
Fig. 12. 


Plate 9. 
Fig. 12. 


Plate 9. 
Fig. j2. 


CHAPTER III. 

Of the Inferior Planets , Mercury and Venus. 

Def. XLIII. The Elongation of any planet is its apparent distance from the 
sun. 

Def. XLIV. The Nodes of the orbit of a planet are the two points, in which the 
orbit cuts the plane of the ecliptic; and a right line, drawn from one node to the 
other, is the Erne of the Nodes. 

Def. XLV. The Limits of the orbit of a planet are two points in the middle be¬ 
tween the two nodes. 

Def. XLVI. An inferior planet is in its inferior conjunction, when it is nearer 
the earth than the sun is, and in its superior conjunction, when it is farther than the 
sun is from the earth, and both in the same secondary. 

Let A be the place of the earth in its orbit ABO, E the place of Venus in its orbit EHG, S the sun, 
and F.D an arc of a circle in the heavens. Venus will be in its inferior conjunction when it is at E, 
and in its superior when it is at G, and both are in the same secondary. 

PROP. LII. An inferior planet is at its greatest elongation, when a line drawn 
from the earth through the planet is a tangent to the orbit of the planet. 

When the planet is at E, being in conjunction with the sun, it has no elongation. As it moves from 
E toward X its elongation increases, till at X, when AF is a tangent to the orbit of Venus, its apparent 
place is F, and its elongation FQ, which is the greatest elongation it can have; for in passing from X to 
G, its elongation decreases, till at G it becomes nothing. This will be true in elliptical as well as 
circular orbits. 

Cor. If the orbits of these planets were circular, the distance of each, from the sun would be to the 
earth’s distance, as the sine of its greatest elongation to the radius. 

The orbits are not circles, but ellipses, having the sun in one focus ; for on this supposition their 
computed places are always found to agree with their observed places. 

PROP. LIII. The inferior planets are never in opposition to the sun. 

For in opposition the earth is between the sun and the planets, which can never happen when the 
orbit of the planet is nearer to the sun than that of the earth. 

Def. XLVII. A planet is in Quadrature , when it is 90 degrees in the celestial 
sphere distant from the sun. 

PROP. LIV. The inferior planets are never in quadrature. 

The greatest angle of elongation is that contained by AQ, drawn from the earth through the sun, 
and AFa tangent to the orbit of the planet. Now if QAF were a right angle, AF would be (El. HI. 
18.) a tangent to the earth’s orbit; but AF is a tangent to an orbit less than that of the earth; it there¬ 
fore makes an angle with AQ, less than a right angle ; that is, QF, the greatest elongation, is less than 
90 degrees. 

Con. Hence the inferior planets never appear far from the sun. 

PROP. LV. While Venus is moving from the superior conjunction to the inferior, 
it sets after the sun ; while it is moving from the inferior conjunction to the superior, 
it rises before the sun. 

Whilst Venus is moving from G, its superior conjunction, through P, to E, its inferior conjunction, 
being in the eastern part of its orbit, the sun will be westward of Venus; therefore Venus, if far 
enough from the sun, will be seen in the west after the sun is gone down, whence it is then called the 
evening star. But on the western side of its orbit, the sun being eastward of it, Venus will set before 
the sun, and consequently rise before it, whence it is then called the morning star. 

PROP. LVI. The greatest elongation of an inferior planet on one side of the sun 
is not always equal to that on the other. 


Chap. III. 


OF THE INFERIOR PLANETS. 


179 


For since the planet moves in an elliptical orbit, at the time of its greatest elongation on one side 
it may be in its aphelion ; and at its greatest elongation on the other side, it may be in some part 
nearer the sun; hence its real distance from the sun at its elongations being unequal, its apparent 
distances will be so likewise. 

Cor. The greatest elongation of Venus being found greater than that of Mercury, the latter must 
be nearer the sun than the former. 


PROP. LVII. The apparent velocity of the inferior planets is greatest at the 
conjunctions. 


Since the plane of the orbit of Venus is oblique to that of the earth, those parts of this orbit, which p| a ( e <>. 
are viewed by a spectator on the earth directly, would appear longer than other equal parts viewed Fig. 12. 
obliquely; whence its motions, if uniform, must appear unequal. If the orbit EPGH of Venus be seen 
obliquely by an eye placed at A, the parts about E and G, or near the conjunctions, will be seen 
directly, for AE is perpendicular to a tangent at E; but the parts about X and P will be seen oblique¬ 
ly ; whence the Proposition is manifest. 

Schol. The time when an inferior planet will come again into a given situation with respect to the 
sun and the earth may be thus found. Whilst Venus performs one revolution, the earth, whose period¬ 
ical time is longer than that of Venus, will not have completed its revolution. Before Venus and the 
earth can be again in the inferior conjunction, Venus must, therefore, besides its entire revolution, de¬ 
scribe an arc equal to that which the earth has passed over; consequently, the number of degrees passed 
overby each, or their angular motions, in the same time, will be reciprocally as their periodical times; 
that is, as the periodical time of the earth is to the periodical time of Venus, so is the angular motion of 
Venus (which is equal to four right angles added to the angular motion of the earth between two in¬ 
ferior conjunctions) to the angular motion of the earth in the same time ; whence (El. V. 17.) as the 
difference between the periodical times of the earth and Venus, is to the periodical time of Venus, so 
are four right angles, or 360°, to the number of degrees over which the earth passes in her orbit from 
one inferior conjunction to another. This is only true upon the supposition that the planets moved in 
circular orbits, in which case the following general rule would apply to the finding of the time from 
conjunction to conjunction, or from opposition to opposition of any two planets. u Multiply their periodic 
times together, and divide the product by their difference, and you have the time sought.” For let P 
= the periodic time of the earth, p = that of the planet (suppose an inferior), t = time required ; then 
360® 30 qo 

P : 1 day :: 360° : -----, the angle described by the earth in 1 day ; for the same reason ——is the angle 


desci'ibed by the planet in 1 day; hence- 

P 


360° 360°. 


is the daily angularvelocity of the planet from the 


earth. Now if they set out from conjunction, they will return into conjunction again, after the planet 

360° 360° p p p p 

has gained 360°; hence--: 360° :: 1 day : t = —. For a superior planet t = -rp. 

° p P P —p P —P 


Def. XLVIII. The apparent motion of a planet, if seen from the earth, is called 
its Geocentric Motion ; if seen from the sun, its Heliocentric Motion. 

PROP. LVIII. When the inferior planets are passing from their stationary point, 
through their superior conjunction, to their stationary point on the other side, their 
geocentric motion is direct, or they appear to move from west to east. 

When Venus is at X, it appears to a spectator at A to be in the point F of the concave sphere of p | ate 
the heavens ; when it has moved forward in its orbit to H, G, N, P, it will appear successively Fl ^' 1 
in the several points F, Q, R, D. But the motion from F to D is from west to east; for whilst the 
sun and earth are on the same side of the planet, it must appear to move in the same direction, 
whether it be viewed from the earth or the sun, because the spectator at either station views the 
concave side of the planet’s orbit; but from the sun it is always seen to move from west to east; 
therefore its apparent geocentric motion in this situation is direct, or in consequentia. 

PROP. LIX. While the inferior planets are moving from the stationary point on 
one side, to the stationary point on the other, through their inferior conjunction, their 
geocentric motion is retrograde, or from east to west. 


tv cc 







3 80 


OF ASTRONOMY. 


Rook VII. Part. I. 


While the planet is in this situation, the convex side of its orbit is toward a spectator on the earth, 
but its concave side toward a spectator at the sun ; hence the former will see the planet move in 
a direction contrary to that in which it will appear to the latter to move. Thus, when the planet 
is at P, it will appear in the heavens at D, and as it passes through E to X, it appears to move from 13 
through R, Q, P, to F; but the motion from D to F is from east to west; therefore the apparent 
motion of the planet in this part of its orbit is relrogade, or in antecedcntia. 

PROP. LX. When the inferior planets are at their greatest elongation, they have 
the same apparent motion as the sun. 

For if at any time a planet be moving apparently slower or faster than the sun, its elongation must 
evidently be increased or diminished, and of course cannot then be the greatest. 

Sciiol. As seen from the sun the motion of these, and indeed of all the planets, appears to be 
constantly direct; the appearance of the stations and retrogradations is entirely owing to the situation 
and motion of the earth. And from these causes the apparent motion of Mercury and Venus appears 
to us to be in looped curves. In order to represent the apparent paths of these planets, Ferguson 
removed from an Orreiy the balls representing the Sun, Mercury, and Venus, and put in their places 
black-lead pencils, with the points upward, and fixed a circular piece of pasteboard, so that it remained 
constantly in the same position during the motion of the machine, with its centre resting on the terres¬ 
trial ball. Then, the pasteboard being gently pressed on the pencils, and the pencils representing 
Mercury and Venus, together with the terrestrial ball, being made to revolve about the pencil in 
the sun’s place, by turning the winch of the machine, the three pencils described on the pasteboard 
Plate 14. n diagram, of which Plate 14, Fig. 1. is a copy, reduced to a less size. - T^e solar pencil marked the 
t ig. l. dotted circle of months, which represents the sun’s apparent motion in the ecliptic, and is the same 
every year. The pencil in Mercury’s place traced the curve, having 23 loops and crossing itself 
the same number of times, which represents that planet’s apparent motion for 7 years. The pencil 
in the place of Venus described the curve having 5 loops, which represents the apparent path of 
Venus for 8 years; after which time this planet moves again very nearly in the same apparent path. 
The ecliptic encloses these curves, in the figure, and dotted lines are drawn from the earth to the 
ecliptic, to show the apparent-or geocentric motion of Mercury for one year, which is easily traced 
from A, where it commences, through B, C, &c. At B Mercury is stationary, and the dotted line, 
drawn through B, shows its place in the ecliptic; from B to C it is retrograde ; at C stationary ; from 
C it proceeds directly. And universally, on each side of the loops the planets appear to be station¬ 
ary ; in the part of the loops nearest to the earth, retrograde; and in the rest of the path, direct. 

PROP. LXI. The heliocentric latitude of an inferior planet is the greatest when 
ille planet is in one of its limits. 

For the planet is then (Def. XLV.) at its greatest distance from the ecliptic, and, therefore, will 
have the greatest latitude, as seen from the sun. 

PROP. LXII. The geocentric latitude of an inferior planet is directly as its heli¬ 
ocentric latitude, and inversely as its distance from the earth. 

The apparent length of a line drawn from the planet to the plane of the ecliptic, that is. its 
geocentric latitude, is (by Book VI. Prop. LXIX. LXX.) directly as its real length, and inversely as 
I lie distance of the spectator’s eye. But the real length of a line drawn from the planet to the plane 
of the ecliptic, is its heliocentric latitude ; and the spectator’s eye is at the earth; whence the prop¬ 
osition is manifest. 

Cor. When Venus is in its inferior conjunction, its heliocentric latitude is less than its geocentric; 
for it is then farther from the sun than from the earth. The contrary takes place with respect to 
Mercury. 

PROP. LXIII. The sun enlightens only one half of a planet, and only one half of 
a planet is visible at once. 

This is sufficiently manifest from the spherical form of the planets, and the rectilinear motion of 
light. 

Schol. Venus and Mercury, in passing from the inferior to the superior conjunction, are observed 
to have all the phases of the moon from new to full. 

Def. XLIX. The hemisphere of a planet which is toward the earth is called its 
Disk, because it appears like a plane circle. 


Chap. IV. 


OF THE INFERIOR PLANETS. 


181 


: 


PROP. LXIV. The inferior planets are invisible in their inferior conjunction ; 
their whole disk is illuminated, when they are in their superior conj inetio i ; and 
they are more or less illuminated, as they are nearer err farther from their superior 
conjunction. 

When Venus, or Mercury, is in its superior conjunction, the whole of its enlightened hemisphere 
is toward the earth, and its entire disk is visible ; as it passes toward its inferior conjunction, its en¬ 
lightened hemisphere turns, by degrees from the earth, till, at the inferior conjunction, it is wnoliy 
turned from the earth, and the planet becomes invisible. 

PROP. LXV. If an inferior planet is in one of its nodes at the time of its in¬ 
ferior conjunction, it will appear as a spot in the disk of the sun. 

When the planet is in the nodes, it will be in the plane of the ecliptic ; and if at the same time 
it be in its inferior conjunction, it will neither appear above nor below the sun, as it does when in 
conjunction in other parts of its orbit, but on the sun’s disk. 

* 


CHAPTER IV. 

' V | nff 2 

Of the Superior Planets , Mars , Jupiter , Saturn , and the Herschel. 

Def. L. The superior planets are Mars, Jupiter, Saturn, and the Herschel. 

PROP. LXVI. The superior planets are sometimes in conjunction with the sun, 
sometimes in opposition, and sometimes in quadrature. 

This is known from observation. 

Let S be the sun; QPO a part of the orbit of Jupiter; P the planet; a d g, or nkg, the earth’s Plate 10. 
orbit. When the earth is at d , the sun at S, and the planet at P, the planet is in conjunct on ; when Fig 1 
the earth is at k. the sun at S, and the planet at P, the planet is in opposition ; when the earth is at 
n or g, and the planet at P, the planet will be in quadrature. 

Con. Therefore the superior planets move in orbits farther distant from the sun, than the orbit 
of the earth. 

PROP. LXVII. The apparent diameter of a superior planet is greatest when the 
planet is in opposition. 

For. when the planet is in conjunction, its distance from the earth is greater than when it is in 
opposition, by the diameter of the earth’s orbit. , 

PROP. LXVIII. If a superior planet were at rest, its apparent geocentric motion 
in any given time would he proportional to the angle subtended by the arc described 
by the earth in that time, and seen from the planet. 

When the earth is at a, the planet P appears in the right line a PA, and among the stars in the Plate 1C*, 
heavens at A; when the earth is at 6, the planet appears at B. Therefore while the earth moves Fig. 1. 
from a to 6, the planet appears to move from A to B. But this arc AB is proportional to the angle 
\PB, that is, to the opposite angle a P 6, which is the angle which the arc a b would subtend to an 
eye placed at the planet P. 

PROP. LXIX. Tiie geocentric velocity of a superior planet is greatest at its 
conjunction and opposition. 

\rcs of a given length near the points d and A-, when the planet is seen from-the earth in con¬ 
junction or opposition, would appear greater than arcs in any other part of the earth’s orbit, viewed 
from the planet P, because the former are seen directly, the latter obliquely ; consequently, these 
arcs would subtend greater angles; whence the apparent velocity of the planet, as viewed from the 
earth, is greater at the conjunction or opposition, than at any other time. 



182 


OF ASTRONOMY. 


Book. VII. Paiit I. 


PROP. LXX. When a line drawn from a superior planet to the earth is a tan¬ 
gent to the earth’s orbit, the planet has the same apparent motion, as it lias viewed 
from the sun. 

At the time described, the earth viewed from the planet is at its greatest elongation from the sun 
(by Prop. Lll.), and therefore has the same apparent motion with it (by Prop. LX.). But the appa¬ 
rent motion of the earth or sun viewed from the planet is equal to that of the planet viewed from 
either the earth or sun, which consequently must be the same. 

PROP. LXXI. When a superior planet is passing from one of its stationary 
situations to the other through the conjunction, its geocentric motion is direct; when 
through the opposition, retrograde. 

While the earth is moving from a through d to g, the sun and planet being both on the same side of 
the earth, the motion of the earth will produce an apparent motion in the sun and planet the same 
way, and both will appear to move from A toward G. But while the earth moves from g to n through 
k , the sun and planet being on contrary sides of the^earth, the motion of the earth in its orbit wiil produce 
an apparent motion of the sun and planet in contrary directions; that is, whilst the sun appears to move 
from west to east, the planet will appear to move from east to west in the direction G, H, I, &c. 

PROP. LXNII. Mars sometimes appears round, sometimes gibbous; Jupiter, 
Saturn, and the Herschel, always appear round. 

When Mars is either in opposition or conjunction, his whole illuminated hemisphere is toward the 
earth, but when he is in quadrature, some part of his illuminated disk is turned from the earth. The 
same must happen in the revolutions of Jupiter, Saturn, and the Herschel, about the sun; but their 
great distance renders the difference between the perfect and imperfect illumination of their disk 
imperceptible. 

Schol. The followingparticulars respecting Mars are given by Dr. Herschel after long and accurate 
observations. 

The axis of Mars is inclined to the ecliptic 59° 42'. 

The node of the axis is in 17° 47' of Pisces. 

The obliquity of the ecliptic on the globe of Mars is 28° 42'. 

The point Aries on the martial ecliptic answer to our 19° 28' of Sagittarius. 

The tigure of Mars is that of an oblate spheroid, whose equatorial diameter is to the polar one as 
1355 to 1272, or as 16 to 15 nearly. 

The equatorial diameter of Mars, reduced to the mean distance of the earth from the sun, is 9 f 8'''. 

And that planet has a considerable, but moderate atmosphere, so that its inhabitants, probably, enjoy 
a situation, in many respects, similar to ours. Phil. Trans, vol. Ixxiv. Part 2. 


As tTie most important particulars respecting the stationary positions of the planets, being erroneous, 
have been expunged from the preceding propositions in this edition, it is thought proper to introduce 
the following. 

Pjiob. To determine the position of two planets when they appear mutually stationary , supposing them to 
move in circular orbits and in the plane of the ecliptic , 

Plate 16. Let 0 be the sun, I the inferior, and S the superior planet. Now as no two planets can ever be 
l'ig. 4. moving in the same right line, it is very obvious that whenever two appear mutually stationary, (or, 

which is the same thing, either appears stationary from the other.) while that appearance continues, a 
light line passing through their centres must be so moved as to continue parallel to itself. Let IS be 
that right line, and v V a right line passing through the same two planets after a certain portion of 
time, taken so small that the parts of the orbits described in the interval may not differ sensibly from 
right lines; if the planets appear mutually stationary the line v V must be parallel to IS, and the lines 
1 i and S s drawn perpendicular to them must be equal. Now taking 1SV from the right angles IS s and 
0 SV respectively, VS s = © SI or the elongation of I from the sun, and in the same manner it may 
be shown that v I i = the supplement of the elongation of S from the sun. But Ir and SV express the 
cotemporaneous motion of I and S respectively, and they are also the secants of i I -a and s SV respect¬ 
ively, If or Ss being radius. Therefore as the velocity of the inferior is to the velocity of the superi¬ 
or, so is the secant of the elongation of the .superior, to the secant of the elongation of the inferior; 



Chap. V. 


OF THE SUPERIOR PLANETS. 


183 




anti since secants arc inversely as the cosines of the same angles, as the velocity of the inferior is to the 
velocity of the superior, so is the cosine of the elongation of the inferior to the costne of the elonga¬ 
tion of the superior. Now from the triangle ©is it is plain that, at all times, the sine of the elonga¬ 
tion of I is to that of S, as the distance of l from the sun is to that of S. Therefore as the relative 
distances are given, and the velocities easily computed from the distances and periods, which are like¬ 
wise given, the problem is reduced to this, viz. The ratio of the sines and the ratio of the cosines 
of two angles being given, to tind those two angles. Put d — the distance, v = the velocity, p —■ the 
periodical time, and o — the circumference of the orbit of the inferior planet, and D, V, P, & O, for 
the same particulars of the superior planet respectively- Let acb and ace (tig. 5.) be the two angles, 
a 6, d e, their sines, and a c, d c, their cosines respectively, radius being 1 ; then d: D :: abide, and 

J) £ 

v : V :: ac:d c. Put ab — x, then d e = —, a c = v/i x 2 . and cd . 






d 


D 2 x 2 . 


, a c = v/l — x 2 , 
D 2 


si 


D 2 x 2 


d 2 


- ; now by substitution 


and v 2 : V 2 : : 1—x 2 : 1 


Now to express the velocities in terms 


d 2 d* 

ot the distances, we find (B. 2. p. 7.) that in uniform motions, the times are as the spaces divided by the 


velocities, that is, P :p 


D 2 

V 2 

1 


d 2 


0 _ 

V 


-, and since circles are as their radii, P :p 


~ 2 ~ ’ but in the solar system P 2 :p 2 :: D 3 : d 3 , therefore D 3 : d 3 :: 


_D 
: "W 
D 2 

"v 2 " 


d 2 


and P 2 :p 2 
V 2 


or D : d 


v 2 ::v '■ ^ 2 ? an d taking the first of these couplets for the last in the last analogy containing x , we 


have D 


: d :: 1 — x 2 : 1 


D 2 x 2 
d 2 ’ 


which by reduction gives x or a b = 


•v/L>2 J.D d r d2 


= sine of 


elongation of the inferior planet, and de = — . ■■■ - = sine of supplement of elongation of the 

v/D2 -f Dd-t d2 11 

superior planet. That is, in words, add together the squares of the distances of any two planets from 
the sun and their product, divide the distance of either by the square root of the sum, and the quotient 
wifi be the natural sine, radius being 1, of the elongation of that planet from the sun, when it appears 
stationary seen from the other. 

This relation between the distances, velocities, and angles does not strictly subsist for any assignable 
time, but as it is gradually approached, and gradually destroyed, it apparently continues for a considera¬ 
ble time, especially with reference to the superior planets. 


CHAPTER V. 

Of the Moon. 

SECTION I. 

Of the Variations in the Appearance of the Moon. 

6ef. LI. When the moon is at its greatest distance from the earth in its orbit, 
which is elliptical, or at its higher apsis, it is said to be in its Apogee ; when at its 
least distance, or lower apsis, in its Perigee. 

Def. LII. When the moon is in conjunction with the sun, it is New Moon ; when 
in opposition, it is Full Moon ; its conjunction and opposition are called by the 
common name of its Syzygies. 

Def. LIII. A Periodical Month is the time in which the moon describes its orbit: 
a Synodical Month is the time which passes between one new moon and the next. 

PROP. LXXIII. A synodical month is longer than a periodical month. 















184 


OF ASTRONOMY. 


Book VII. Part L 


Plate 10 
Fig. 8. 


Plate 10 
Fig. 2. 


Plate 10. 
Fig. 3. 


Plate 10. 
Fig. 3. 


Because the moon moves in the same direction with the sun, while the moon performs one revo¬ 
lution in its orbit, the sun, by its apparent annual motion, is advanced in the ecliptic; consequently 
the moon must pass beyond the point in which it has completed its revolution before it comes again 
into conjunction with the sun. 

Let S be the sun, BA a part of the earth’s orbit, m d, MD, the diameter of the moon’s orbit when 
the earth is at B, or A. While the earth is at A, if the moon be at D, it will be in conjunction, and 
if the earth continued in the same place, after one revolution in its orbit, it would be again in con¬ 
junction; but if, during the revolution of the moon, the earth is removed to B, the moon at the end of 
the revolution will be at d, a point which is not between the earth and sun; it must therefore move 
on from cl to e before it will be in conjunction. 

PROP. LXXIV. The moon, at its conjunction, is invisible ; at its opposition, its 
whole disk is enlightened ; at its quadrature, it is half enlightened ; between the 
conjunction and quadrature, it is horned; and between the quadrature and oppo¬ 
sition, it is gibbous. 

Let QTL be a part of the earth's orbit, S the sun, T the earth, ACEG the moon’s orbit. When 
the moon is at E, or in conjunction, because it.is between the earth and sun, its illuminated hemi¬ 
sphere will be wholly turned from the earth, consequently its disk will be dark. At A, being in opposi¬ 
tion, its illuminated hemisphere will be wholly toward the earth, and its whole disk will be visible. 
At C, or G, the apparent distance of the moon from the sun will be 90 degrees ; for a right line 
from C to G will make a right angle with the line TS, in which the sun appears; whence the moon 
at each of them is half way between the opposition and conjunction, that is, in the middle slate be¬ 
tween the perfect illumination and the entire darkness of its disk; consequently, its disk is half 
enlightened. In passing from C to E, and from E to G, its disk will be less than half illuminated, 
or it will appear horned; in passing from G to A, and from A to C, its disk will be more than half 
illuminated, or it will appear gibbous. 

Cor. The earth mtist be a satellite to the moon, and subject to the same changes, but more than 
13 times larger than the moon appears to us. At new moon to us the earth appears full to her. 

Def. LIV. A circle supposed to be drawn upon the surface of the moon, separat¬ 
ing the illuminated from the dark hemisphere, is called the Circle of Illumination; 
a circle which separates its visible from its invisible hemisphere, is called the Circle 
of the Disk. 

PROP. LXXV. If the centres of the sun, the earth, and the moon, are joined 
by straight lines forming a triangle, the external angle of this triangle at the moon 
is equal to the angle contained between the circle of illumination and the circle of 
the disk. 

Let S be the sun, T the earth, mrnq the moon, aud O its centre. Let STO be the supposed 
triangle. Draw the line r q perpendicular to SO, and nm perpendicular to TOP. The angle SOP 
will be equal to the angle in O q, contained between r q, which represents the circle of illumination, 
and nm, which represents the circle of the disk. Because the angles SO q, PO m, are by construction 
right angles, they are equal; taking SO m from each, the remaining angles POS, m O q, are equal. 
Consequently the remaining angles POS, m O (/, ai'e equal. 

Schol. This Proposition serves to explain all the different phases of the moon. For example ; 
when the moon is moved from O to A, the line SO coincides with SA, and TO with TA; therefore 
TO, OS, lie in the same line, and the external angle POS is nothing; whence the two circles of 
illumination and of the disk coincide ; and because the disk is then turned from the sun, it is wholly 
dark. When the moon is in quadrature, the line SO will be a tangent to the moon’s orbit; whence 
SOP will be a right angle, and the two circles will be at right angles to each other, and the disk 
will appear half illuminated. If the angle POS be less or greater than a right angle, the circle of 
illumination will make an angle with that of the disk less or greater than a right angle ; whence the 
illuminated part will appear horned or gibbous. Lastly, when the moon is in opposition, the lines 
SO, ST, become one and the same line ; whence the circles coincide, and the whole disk is illuminated. 

PROP. LXXVI. The horns of the moon, after the conjunction, are turned toward 
the east: before it. toward the west. 


Ciiap. V. 


OF THE MOON. 


185 


The sun, after the conjunction, setting before, that is, to the west of the moon, illuminates the west 
side of the moon’s disk; whence its horns, which are toward the dark part of the disk, are toward 
the east. The reverse takes place, before the conjunction, when the moon is seen in the east, before 
the sun- rises. 

PROP. LXXVII. When the moon is horned, its obscure part is visible by the 
reflection of.the rays of the sun from the earth. 

Wnen the moon is at D or F, near the conjunction, the enlightened hemisphere of the earth will pi a te 10. 
be toward the moon, and reflect the rajs of the sun upon it. Fig. 2. 

PROP. LXXYI1I. The moon always has nearly the same side toward the 
earth. 

Th s is proved by observation. 

Cor. 1 . Hence, if the moon revolves about its axis, its periodical time must be equal to that of 
its -revolution in us orbit round the earth. This is also found to be the case with the fifth satellite of 
Saturn as it regards its primary. 

Cor. 2. Though the year is of the same absolute length, both to the earth and moon, yet the 
number of days in each is very different; the former having 365^ natural days, but the latter only 
about 12JL, every day and night in the .moon being as long as 29 \ on the earth. 

Schol. This proposition would be true without any limitation, if the angular velocity of the moon 
about the earth were equal to the angular velocity about her axis, for then the same face must be 
always exactly turned toward the earth. But the angular velocity of the moon about the earth is 
not uniform, while that about its axis is uniform, and therefore the same face is not always turned 
to the earth. Since, however, the opposite face of the moon is never seen, the time of the moon’s 
rotation about her axis must be equal to the mean time in her orbit. See Prop. LXXX. and LXXXI. 

PROP. LXXIX. The moon appears to have two librations, one upon a line per¬ 
pendicular to its axis, called its libration in latitude ; the other upon its axis, called 
its libration in longitude. 

This appears from observation, some small portions of the surface of the moon being visible in 
some parts, and invisibe in other parts, of its orbit; that is, in consequence of her libration in latitude 
we sometimes see one pole, and sometimes the other. And by the libration in longitude, more of 
the western limb is sometimes seen, and at others, more of the eastern. The inclination of her axis 
to her orbit is 6° 49'. 

Cor. The axis of the moon being almost perpendicular to the ecliptic, there will be but a small 
difference in its seasons.. 

Schol. The moon’s axis being so nearly perpendicular to the ecliptic, that the small obliquity of 
her orbit cannot cause the sun sensibly to decline from the equator; hence we learn, that the inhab¬ 
itants of the moon must devise some method, different from ours, for ascertaining the length of their 
year. We know the length of our years by the return of the equinoxes; but these are not discover¬ 
able in the moon, where the days and nights are equal. 

The Lunarians may, perhaps, measure their year, by observing when either of the poles of our 
earth begins to be enlightened, and the other to disappear, which happens at our equinoxes; they 
being conveniently situated for observing large tracts of land about the poles of the earth, which, at 
present, are unknown to the most experienced of our nav igators. 

PROP. LXXX. The librations of the moon may be explained on the supposition 
that the moon has a revolution round its axis. 

Let IH represent the plane of the moon’s orbit, E the earth, and C3ID the moon ; and let AB be pi a te 10 
the axis round which the moon revolves, and A be called its north pole, and B its south pole. CMD Fig. 4, 
will in this situation be the visible hemisphere, and CD the plane of the disk. By the libration in 
latitude the line AB appears to vibrate on the line IH, so that sometimes the point A is visible, and 
sometimes the point B. This variation attends the moon’s revolution in its orbit fin one half of the 
orbit the pole A is always visible, and in the other half the pole B. It must therefore arise from the 
inclination of the axis of the moon to the plane of its orbit. When the moon is at I, if the axis AB 
be inclined to III, the plane of the orbit, making with it the angle AIH to a spectator at E, the visible 
hemisphere will be CisD, and the pole A will be within the disk; but when the moon is at H, the 

24 


186 


riate 10. 
Fig. 5. 


late 10 
Fig. 4. 

Fig. 5. 

Fig. 2. 


Tlate 14 
Fig. 2. 


OF ASTRONOMY. Book VII. Part I. 

visible hemisphere will be CMD ; and the axis AB being always parallel to itself, the pole B will be 
within the disk. 

Again, let the moon be at A, the earth at T. If a circle, whose plane is perpendicular to T c, pass 
through the line dcs , this circle will be that of the disk, and dbps will be the visible hemisphere. 
But the moon has an apparent motion, by which the disk is changed, so that at one time rb is the 
circle of the disk, rpb the visible hemisphere, at another fp and fcp. In the former case, sr becomes 
visible, and d b invisible; in the latter case, df becomes visible, and p s invisible. This libration in 
longitude arises from the elliptical form of the moon’s orbit. If the moon has a rotation round its 
axis, it has been shown that its revolution will be completed in the time of one revolution in its orbit; 
and because the motion round the axis is uniform, one quarter of a revolution will be completed in 
one fourth part of the periodical time. But, the moon’s orbit being supposed elliptical, and the earth 
placed at T one of the foci, the moon will move slower at A its apogee, than at P its perigee, and 
its velocity will continually increase in moving from A to P, and decrease in moving from P to A. 
If therefore when the moon is at A, d bps is the visible hemisphere, after it has moved from A to B, 
through that quarter of its orbit, in which it moves with its less velocity, and consequently takes up 
more than a quarter of its periodical time, ds will not be perpendicular to Tc, but the point s will 
have turned from west to east more than a quarter round the axis C; hence the point s will not be 
visible when the moon is at B, and fp instead of ds will be the circle of the disk. In passing from 
B to P, its excess of velocity will make up for its defect in passing from A to B ; and at B it will 
have completed half its orbit in half its periodical time ; but in half its periodical time, it will have re¬ 
volved half round its axis, therefore at P, ds will again be perpendicular to T c, and dxs will again 
be the visible hemisphere. The reverse will take place in passing from P to A through D, when br 
will be the circle of the disk, and s will be within it. 

PROP. LXXNI. The moon revolves about its axis in the same time in which it 
revolves about the earth. 

Without such a revolution, the phenomena of its librations could not happen. If the point A 
were visible in one part of the moon’s orbit, it would be always visible, without a rotation about 
an axis, oblique to the plane of the orbit, to produce an apparent motion of the point A, or the libra¬ 
tion of latitude. If ds were perpendicular to c T, when the moon is at A, it would be so in every 
other part of the orbit; and therefore dxs would always be the illuminated hemisphere, if there were 
not a revolution about the axis to produce, in the manner above explained, the libration of longitude. 
These librations therefore prove the existence of this revolution; and it has been shown, that if there 
be such a revolution, its periodical time is the same with that of the moon in its orbit. 

Cor. Hence it is evident, that one half of the moon is never in darkness; the earth (Prop. 
LXXIV. Cor.) constantly affording it a strong light, during the absence of the sun; but the other half 
has a fortnight’s light and darkness by turns. 

PROP. LXXXII. The orbit of the moon is an ellipse. 

It is only on this supposition that the libration of longitude can be explained ; from tiffs phenome¬ 
non, therefore, the elliptical form of the orbit may be inferred. 

Schol. The moon revolves about the earth in an ellipse, and with the earth revolves about the sun. 
In Fig. 2, Plate 14, the broad curve line ABODE represents as much of the earth’s annual orbit, as it 
describes in 32 days, which are marked with numeral figures. The small circles at «, b , c, ff, e, show 
the moon’s orbit in proportion to that of the earth, and the narrow curve ab cdef represents the 
moon’s path in the heavens, or with respect to the sun, for 32 days, reckoning from a new moon at a. 

The earth being supposed to move from A, and the new moon at the same time from a ; 7 days 
afterward the earth will be at B, and the moon in quadrature at b ; when the earth is at C, the moon is 
full at c; when the earth is at D, the moon is in quadrature at d; and when the earth is at E, the 
moon is again new, or in conjunction at e. 

From this figure it appears, that the moon’s path is alwa} ? s concave toward the sun. For, if we 
suppose the concavity of the earth’s orbit to be measured by the length of a perpendicular C g, falling 
from the place of the earth at C, when the moon is full, on the right line bgd, connecting the places 
of the earth at the end of the first and third quarters of the moon, the length of the perpendicular will 
be about 640,000 miles; when the moon is new, it is about 240,000 miles nearer to the sun than the 
earth is; therefore the length of a perpendicular from its place at that time on the same right line 
will be about 400,000 miles, and shows the concavity of that part of the path. 

PROP. LXXXIII. The moon’s surface is irregular. 

If the surface were every where regular, the limit between the enlightened and dark parts of the 
disk being the circle of light, that is, a perfect great circle of a sphere, would be exactly defined when 
the moon is horned, half enlightened, or gibbous, contrary to observation. 


Chap. V. 


OF THE MOON. 


187 


Scuoc. The face of the moon, as seen through^ telescope, appears diversified with hills and vallies. 
This is proved by viewing her at any other time than when she is full; for then there is no regular line 
bounding light and darkness; but the edge or border of the moon appears jagged; and even in the dark 
part near the borders of the lucid surface there are seen some small spaces enlightened by the sun’s 
beams. 

Besides, it is moreover evident, that the spots in the moon taken for mountains and vallies, are 
really such from their shadows. For in all situations of the moon, the elevated parts are constantly 
found to cast a triangular shadow in a direction from the sun ; and the cavities are always dark on the 
side next to the sun, and illuminated on the opposite side. Hence astronomers are enabled to find the 
height of the lunar mountains. Dr. Keil, in his Astronomical Lectures, has calculated the height of 
St. Catharine’s hill to be nine miles. Since, however, the loftiest mountains upon the earth are but 
about three miles in height, it has been long considered as very improbable that those of a planet so 
much inferior in size to the earth should exceed in such vast proportion the highest of our mountains. 

By the observations of Dr. Herschel, made in November, 1779, and the four following months, we 
learn, that the altitude of the lunar mountains has been very much exaggerated. His observations 
were made with great caution, by means of a Newtonian reflector, 6 feet 8 inches long, and with a 
magnifying power of 222 times, determined by experiment; and the method which he made use of to 
ascertain the altitude of those mountains, which, during that time, he had an opportunity of examining, 
seems liable to no objection. The rock situated near Lacus Niger , was found to be about one mile in 
height, but none of the other mountains, which he measured, proved to be more than half that altitude; 
and Dr. Herschel concludes, that with a very few exceptions, the generality of the lunar mountains do 
not exceed half a mile in their perpendicular elevation. See Keil’s Astron. Lect. x. Phil. Trans. 
Vol. lxx. 

To Dr. Herschel also w r e are indebted for an account of several burning volcanoes, which he saw 
at different times in the moon. In the 77th vol. of the Phil. Trans, he says, April 19, 10 hours 36 
minutes sidereal time, “ I perceive three volcanoes in different places of the dark part of the new moon. 
Two of them are nearly extinct; or, otherwise, in a state of going to break out, which perhaps may 
be decided next lunation. The third shows an actual eruption of fire, or luminous matter.” On the 
next night, Dr. Herschel saw the volcano burn with greater violence than on the preceding evening. 
He considered the eruption to resemble a small piece of burning charcoal when it is covered by a thin 
coat of white ashes, which frequently adhere to it, when it has been some time ignited; and it had a 
degree of brightness about as strong as that with which such a coal would be seen to glow in faint day¬ 
light. It is not yet determined whether there is an atmosphere belonging to the moon. The question 
is fully discussed in the Encyclo. Brit. Vol. II. p. 457, &c. See also Phil. Trans, for the year 1792. 

SECTION II. 

Of Eclipses 

Def. LV. An Eclipse of the moon happens when the earth, passing between the 
sun and moon, casts its shadow on the moon. 

PROP. LXXXIV. The moon can only be eclipsed at the full, or in opposition. 

For it is only when the moon is in opposition that it can come within the shadow of the earth, which 
must always be on that side of the earth which is from the sun. 

PROP. LXXXV. The moon can only be eclipsed when, at the full, it is in or near 
one of its nodes. 

The earth being in the plane of the ecliptic, the centre of its shadow is always in that plane ; if 
therefore the moon be in its nodes, that is, in the plane of the ecliptic, the shadow of the earth will 
fall upon it; also, since this shadow is of considerable breadth, it is partly above and partly below the 
the plane of the ecliptic; il therefore the moon in opposition be so near one of its nodes, that its lat¬ 
itude is less than half the breadth of the shadow, is will be eclipsed. But, because the plane of the 
moon’s orbit makes an angle of more than 5 degrees with the plane of the ecliptic, it will frequently 
have too much latitude at its opposition to come within the shadow of the earth. 

PROP. LXXXVI. The sun is larger than the earth, and the shadow of the earth is 
a cone, the base of which is on the surface of the earth. 

If the earth were larger than, or equal to, the sun, it is manifest, that its shadow would either per¬ 
petually enlarge, or be always of the same dimension; but in this case, the superior planets would 


188 


OF ASTRONOMY. 


Book VII. Part I. 


Plate 10. 
Fig’. 7. 


Plate 10. 
Fig. 6. 


Plate 10. 
Fig. 7- 


\ 


Plate 10. 
Fig. 10. 


Plate 10. 
Fig. 10. 


Fig, 9. 


sometimes come within it, and he eclipsed, which never happens. Therefore the sun is larger than 
the earth, and produces a shadow from the earth of a conical form, which does not extend to the orbit 
of Mars. 

PROP. LXXXVII. The moon is eclipsed by a section of the earth’s shadow. 

Let CDE be the earth, CME the cone of its shadow, and FH a part of the moon’s orbit passing 
through the shadow ; it is manifest that as the moon describes this part of its orbit, it passes through 
the circular section FGHL. 

Def. LVI. The moon’s Horizontal Parallax is the angle which a semidiameter 
of the the earth would subtend, if it were viewed directly from the moon. 

LEMMA. Half the angle of the cone of the earth 1 s shadow may be taken as equal 
to the angle of the sim’s apparent semidiameter . 

Let AFBG be the sun, HED the earth, FIMD or BMA the angle of the cone of the earth’s shadow r , 
and CMD half this angle. SA, a semidiameter of the sun, drawn from its centre to the point of con¬ 
tact of the tangent AM, is perpendicular (El. III. 16.) to AM, and is therefore seen directly from D; 
and it subtends the angle SDA ; it must therefore appear large in proportion to the magnitude of 
this angle. But in the triangle SDM, the external angle SDA is equal to the two angles CSD, CMD ; 
of which, CSD, the angle in which the semidiameter of the earth is viewed from the sun, is so 
small, that without any sensible error it maybe reckoned as nothing, and SDA be said to be equal 
to SMD. 

PROP. LXXXVIII. The semidiameter of the section of the earth’s shadow, 
which eclipses the moon, is equal to the difference between the horizontal parallax 
of the moon, and the sun’s apparent semidiameter. 

CT, being a semidiameter of the earth drawn from the point of contact of the tangent CM, is 
perpendicular to CM. CT will therefore be seen directly from the point F in the moon’s orbit, 
subtending the angle CFT, which is (Def. LVI.) the moon’s horizontal parallax. FMG is the semiangle 
of the cone of the earth’s shadow, equal to the angle of the sun’s apparent semidianieter, because MC 
produced would be a tangent to the circle of the sun’s disk. FG is the semidiameter of the section 
FGHL of the earth’s shadow through which the moon passes in an eclipse ; and FTG the angle which 
this semidiameter will subtend when it falls upon the moon, and is viewed from the earth. Now 
the angle CFT is equal to the two angles FMG, FTG (El. I. 32.), consequently, FTG is equal to 
CFT— FMG ; but by the preceding Lemma, FMT may be taken for the angle of the sun’s apparent 
semidiameter; and CFT is the moon’s horizontal parallax; whence the Proposition is manifest. 

Cor. Hence the section of the earth’s shadow, by which the moon is eclipsed, is broader than 
the moon ; for the semidiameter of the shadow, by this Prop, is 61'—16'= 45', and the diameter 
of the moon is about 31'. 

Def. LVII. An eclipse of the moon is partial , when only a part of its disk is 
within the shadow of the earth ; it is total, when all its disk is within the shadow ; 
and it is central, when the centre of the earth’s shadow falis upon the centre of the 
moon’s disk. 

PROP. LXXXIX. The moon, at the full, will not be eclipsed, if its latitude is 
equal to, or greater than, the sum of its own semidiameter, added to the semidiame¬ 
ter of the earth’s shadow. 

Because a circle or ellipse appears as a right line when the eye is in the same plane with it, let 
00 represent the plane of the ecliptic, RR the plane of the moon’s orbit, and N the node. At the full 
moon, if the earth’s shadow be at A, the moon F must be in the same part of the heavens, because 
it is in opposition. But because only one half of the shadow of the earth, or about 45' is on the same 
side of the ecliptic with the moon, and only one half of the moon’s breadth, or about 16', is on the 
side of its orbit toward the earth’s shadow; if the centre of the moon be more than 61'from the 
centre of the shadow, the moon will pass clear of the shadow, and will not be eclipsed. 

Let GE be an arc of the moon’s orbit, AB an arc of the ecliptic, and C c an arc in a secondary of 
the ecliptic equal to the moon’s latitude. If this arc be equal to, or greater than, C t and c /, the sum 
of the semidiameters of the earth’s shadow, and of the moon, it is manifest that the shadow cannot 
pass over any part of the moon’s disk. N 


Chap. V. 


OF THE MOON. 


189 


PROP. XC. The moon, at the full, will be partially eclipsed, if its latitude is 
less than the sum, but greater than the difference, of its own semidiameter and that 
of the earth’s shadow. 

If Cc, the latitude of the moon, be supposed less than C t, c /, the sum of the semidiameters of the Plate 10 . 
shadow and the moon, /, the lower edge of the moon, will be below t, the upper edge of the shadow ; Fig. 9 - 
whence the side of the moon, toward the ecliptic will be eclipsed. But, because C c, the moon’s 
latitude, added to c o, its semidiameter, is greater than C t, the semidiameter of the earth’s shadow., 
the upper edge of the moon o cannot come within the shadow; whence the eclipse will be partial. 

And because in this case the moon’s latitude, together with its semidiameter, is greater than the se¬ 
midiameter of the shadow, the moon’s latitude is greater than the difference of the semidiameters 
of the shadow and the moon ; or, because Cc -f co is greater than C t, C c is greater than C t — c o. 

PROP. XCI. The moon will, at the full, be totally eclipsed, if its latitude be less 
than the difference between its own semidiameter, and that of the earth’s shadow. 

If C c -f- c o be less than C t, that is, if C c be less than C t — c /, the upper edge of the moon may Fig- 9* 
come within the shadow of the earth. 

PROP. XCII. The moon is centrally eclipsed, only when, in opposition, it is in 
one of its nodes. 

The node N, being the common intersection of the moon’s orbit, and the plane of the ecliptic, is in 
both. Therefore when the moon is in the node, its centre is in the plane of the ecliptic, in which is 
the centre of the earth’s shadow, and consequently at the full these centres coincide. 

Schol. 1. Both the moon and the shadow moving from west to east, the moon would always be 
in eclipse while it was at, or near, its nodes, if it moved with the same velocity as the earth ; but 
because it moves much faster than the shadow of the earth, it soon passes from its opposition out of 
the'shadow. 

Schol. 2. The*eemiangle of the cone of the earth’s shadow being known, the length of the shadow 
may be found. 

The semiangle CMT being equal to the sun’s apparent semidiameter, or about 16', and CT a semi- Plate 10. 
diameter of the earth being perpendicular to the tangent CM, if TM be radius, CT is the sine of the Fig. 7. 
angle CMT. Therefore, as the sine of an angle of 16' is to radius, so is CT to TM, the height of the 
cone; that is, as l to 217. Whence the length of the shadow is about 217 semidiameters of the 
earth. 

PROP. XCIII. The moon in a total eclipse, is not wholly invisible. 

This is known by observation ; and the phenomenon is produced by the rellection of rays of light 
falling upon the earth’s atmosphere, toward the shadow, and consequently toward the moon in the 
shadow. 

Def. LVIII. An Eclipse of the Sun happens when the moon, passing between the 
sun and the earth, intercepts the sun’s light. 

PROP. XCIV. The sun can only be eclipsed at the new moon. 

For it is only when the moon is in conjunction, that it can pass directly between the sun and the 
earth. 

PROP. XCV. The sun can only be eclipsed, when the moon, at its conjunction, is 
in or near one of its nodes. 

For unless the moon is in or near one of its nodes, it cannot appear in or near the same plane with 
the sun; without which it cannot appear to us to pass over the disk of the sun. At every other part of 
its orbit, it will have so much northern or southern latitude, as to appear either abo-ve or below the 
sun. If the moon be in one of its nodes, having no latitude, it will cover the whole disk of the sun, and 
produce a total eclipse , except when its apparent diameter is less than that of the sun; if it he near one 
of its nodes, having a small degree of latitude, it will only pass over a part of the sun’s disk, or the 
eclipse will be partial. 


190 


Plate 10. 
Fig. 11. 


Plate 10. 
Fig. 6. 


Plate 10. 
Fig. 7. 


Plate 10. 
Fig. 11. 


Plate 10. 
Fig. 11. 


OF ASTRONOMY. Book VII. Part I. 

PROP. XCVI. In a total eclipse of the sun, the shadow of the moon falls upon 
that part of the earth where the eclipse is seen. 

Let SL be the sun, TR the moon, VM a part of the surface of the earth. A spectator placed any 
where between V and M, will not see any part of the sun, because the moon will intercept all the rays 
of light which come to him directly from the sun; and it is manifest, that, in this situation, the moon, 
being an opaque body, will cast its shadow upon VM, the part of the earth where the eclipse is total. 

• PROP. XCVII. The shadow of the moon is a cone terminated in a point. 

Because (by Prop. LXXXV1II. Cor.) the diameter of the moon is less than the diameter of the 
earth’s shadow, where it eclipses the moon, and this diameter (by Prop. LXXXVI.) is less than the diame¬ 
ter of the earth, the diameter of the moon is much less than that of the sun ; consequently, its shadow 
will be a cone, terminated in a point. The tangents LAR, SAT, terminate in A ; and only the rays 
that would pass within these tangents are intercepted by the moon; therefore RTA is the form of the 
moon’s shadow. 

LEMMA. Half the angle of the cone of the moon’s shadow is equal to the angle of 
the apparent semidiameter of the sun. 

Let FBGA be the sun ; HED the moon; the cone HMD the moon’s shadow; CMD the semiangle 
of this cone ; SA the semidiameter of the sun, and SDA the angle which this semidiameter would sub¬ 
tend, if viewed from D. It may be proved, as in Prop. LXXXVIII. Lemma , that CMD is equal to SDA ; 
for the distance of the moon from the sun is so nearly equal to that of the earth from the sun, that the 
apparent semidiameter of the sun, as seen from the earth or moon, may be considered as equal. 

PROP. XCVIII. A semidiameter of the moon’s shadow, where it falls upon the 
earth, is equal to the difference between the apparent semidiameters of the moon 
and sun. 

Let CDE represent the moon, CME the cone of its shadow; FG the semidiameter of the moon’s 
shadow where it falls upon the earth in a solar eclipse; CT the semidiameter of the moon, CFT, its 
angle viewed from F, and FTG the angle of the apparent semidiameter of the moon’s shadow' viewed 
from T. 

In the triangle TFM, the external angle TFC (El. I. 32.) is equal to the two angles FTG, FMG 
Therefore FTG is equal to the difference between TFC and FMG; and FMG is (by the preceding 
Lemma) equal to the angle of the sun’s apparent semidiameter, and TFC is the angle of the moon’s 
apparent semidiameter; w'hence the Proposition is manifest. 

PROP. XCIX. In a partial eclipse of the sun, a penumbra , or imperfect shadow of 
the moon, falls upon that part of the earth where the partial eclipse is seen. 

A spectator at N, or P, might see the whole sun ; for a ray passing from the most remote side of 
the sun, S or L, w'ould not be intercepted by the moon. But at any point in NM, VP, the spaces be¬ 
tween the moon’s shadow and the points N, P, the spectator would only see a part of the sun; thus at 
G, or D, he would see that half of the sun which lies without the tangents DRC, GTC ; consequently, 
in all places between the points N, M, and P, V, there will be less light from the sun, than if it were 
not at all eclipsed. This deficiency of light is called the moon’s penumbra. 

PROP. C. The moon’s penumbra is an increasing cone, and its darkness increases 
toward the shadow of the moon. 

While a spectator is in the space between N and M, or P and V, he is in the penumbra; but at 
the points N, P, passes out of it. Therefore the tangents NS, PL, are the limits of the penumbra. 
If tangents be supposed drawn round the spherical surfaces of the sun and moon, they will form two 
cones, having their common vertex at F, and increasing, the one toward the points L, S, the other 
toward the points N, P. And as the spectator moves from N toward M, or from P toward V, a °reat- 
er and greater portion of the sun continually becomes invisible to him ; whence the penumbra increas¬ 
es in darkness toward M and V. 

PROP. CL The semiangle of the moon’s penumbra is equal to the angle of the 
sun’s apparent semidiameter. 


Chap. V. 


OF THE MOON. 


191 


Let SL be the sun, and TR. the moon. By Prop. C. the 'tangents SFN, LFP, terminate the cone Plate 10. 
of the penumbra. CE, drawn from the centre of the sun to that of the moon, bisects TFR, the angle P’S- 11 - 
of this cone; whence EFT is its semiangle. LC being the semidiameter of the sun, LTC is the an¬ 
gle under which this semidiameter would appear from the moon T. I say TFE is equal to LTC. 

For, in the triangle TCF, the external angle EFT is equal to the two internal and opposite angles 
FTC, FCT ; that is, to the two angles LTC, TCE. Therefore EFT is equal to LTC, TCE. But 
TCE, being the angle which the moon’s semidiameter would subtend, if viewed from the sun, is so 
small that it may be neglected; whence EFT may be considered as equal to LTC. 

PROP. CII. The semidiameter of the moon’s penumbra, in that part through 
which the earth passes in an eclipse of the sun, is equal to the sum of the apparent 
semidiameters of the sun and moon. 

Let CD be the moon, CDAB its penumbra, and CMB the angle of the cone of the penumbra; and Plate 10. 
let AEBF be the section of the penumbra through which the earth passes in an eclipse of the sun, Fi S- 12 * 
AB its diameter, and AT its semidiameter. MLT, drawn from the vertex through the centre of the 
moon, will bisect the angle CMD. Therefore (by Prop. Cl.) CML, the semiangle of the penumbra, 
is equal to the sun’s apparent semidiameter. And CL is the moon’s apparent semidiameter, as seen 
from the earth A, subtending the angle CAL; and AT, the semidiameter of the penumbra, seen from 
L, subtends the angle ALT. Now the angle ALT is equal to the two angles CML, CAL ; whence the 
truth of the Proposition is manifest. 

Def, LIX. The disk of the earth is that hemisphere of the earth, which is seen, 
as a circle, from the moon. 

PROP. CIII. At the new moon the whole disk of the earth is enlightened. 

For, the moon being then between the sun and the earth, the earth viewed from the moon will appear 
in opposition, and consequently its enlightened hemisphere will be toward the moon. 

PROP. CIV. The semidiameter of the earth’s disk is equal to the moon's hori¬ 
zontal parallax. 

The moon’s horizontal parallax is the apparent semidiameter of the earth, as viewed from the 
moon, that is, it is equal to the semidiameter of the disk, since the disk is a hemisphere of the earth 
viewed from the moon. 

PROP. CV. If the latitude of the moon, when new, is equal to, or greater than, 
the sum of the semidiameter of the penumbra and the moon’s horizontal parallax, 
there will be no eclipse of the sun; if less, there will be an eclipse either partial, or 
total. 

Let ACB be a part of the ecliptic ; let GNE be the plane of the moon’s orbit; let N be the node, p iatp l0 
DL the earth’s disk, and o l a section of the moon’s penumbra. Because the centre of the moon’s pig. 9. 
penumbra is always in a right line passing through the centres of the sun and the moon, the distance of 
c, the centre of the moon’s penumbra, from the plane of the ecliptic, must always be the same with the 
distance of the centre of the moon from the same plane ; that is, with the latitude of the moon. And be¬ 
cause the centre of the earth, or the earth’s disk C, is in the ecliptic, C c, the distance between the cen¬ 
tres of the penumbra and of the earth’s disk, is always equal to the latitude of the moon. Now if C c, 
or the latitude of the moon, be equal to, or greater than cZ, the semidiameter of the penumbra, 
together with t C, the semidiameter of the earth’s disk, or the moon’s horizontal parallax, then no part 
of the penumbra will fall upon the disk, that is, there will be no eclipse. If C c the latitude of the moon, 
be less than t C + Z c, the edge of the penumbra will be nearer the ecliptic than the edge of the disk, 
and there will be a partial eclipse. And if C c be less than C Z, the shadow of the moon will pass 
over some part of the disk of the earth, and where this happens, the eclipse will be total. 

PROP. CVI. If the moon, when new, is in one of its nodes, the eclipse of the 
sun will be central. 

For then the centres of the earth, sun, and moon, being all in the plane of the ecliptic, the centre 
of the moon will pass between the sun’s centre, and that of the earth. 


/ 


192 OF ASTRONOMY. Rook VII. Part I. 

Schol. 1 . The penumbra of the moon in a central eclipse will not cover the whole disk of the 
earth. 

The semidiameter of the moon’s penumbra, being equal to the sum of the apparent semidiameters . 
of the sun and moon, that is, about 16', 23" -p 15' 37", or 32' at the medium, its diameter is about 64'; 
whereas the diameter of the earth’s disk is about 120 '; whence the penumbra cannot cover the whole 
disk. 

Schol. 2 . The height of the shadow of the moon is about GO* semidiameters ol the earth. The 
semiangies of the earth’s shadow, and of the moon’s shadow, being each equal to the sun’s apparent 
semidiameter, the angles are equal to one another, and these cones are similar. Therefore as the 
semidiameter of the base of the earth’s shadow (that is, of the earth) is to the semidiameter of the 
base of the moon’s shadow (that is, of the moon), so is the height of the earth’s shadow to the height 
of the moon’s shadow. Now the semidiameter of the earth is to that of the moon, nearly as 1 U 0 to 
_ 28, and the height of the earth’s shadow is about 217 semidiameters of the earth; whence the height 

of the moon’s shadow is equal to about 6 l)A semidiameters of the earth ; for 100 : 28 : : 217 : 60A nearly. 

Def. LX An eclipse of the sun is said to be annular , when, at the time of the 
eclipse, a ring of the sun appears round the edges of the moon. 

PROP. CV1I. A central eclipse of the sun will be an annular one, if the distance 
of the moon from the earth at the time of the eclipse be greater than its mean dis¬ 
tance. 

Plate 10 . SL being the sun, TR the moon, TAR the moon’s shadow, and EA the height of this shadow, which 
E'g- 1 R is about 60 £ semidiameters of the earth; if at the time of a central eclipse PJ\ is a part of the surface 
of the earth, those who live in the parts PV, MN, being in the penumbra, will (by Prop. XCIX.) see a 
partial eclipse; and tnose who live between V and M, being in the shadow, will (by Prop. XCV1.) see 
a total eclipse. But if the distance of the moon from the earth be equal to EA, or 60§ semidiameters 
of the earth (which is the moon’s mean distance), A will be the only point from which the eclipse will 
appear total. And if the moon’s distance be greater than EA, as EG, the shadow not reaching the earth, 
there will be no total eclipse. Consequently, though a spectator at O would see a central eclipse 
(because the centres C, E, are in the same line with the point of vision S),yet the eclipse would not be 
total, because the spectator is notin the shadow of the moon. Hence it must appear annular; for let 
ORX be a tangent to the moon drawn from the eye at O; and it will fall upon the sun at X, and the 
part XL of the sun will be visible; in like manner parts of the sun equal to XL will be visible all round 
the moon, forming a ring. 

Cor. Hence it appears, that in an annular eclipse it is the penumbra of the moon, which falls upon 
the earth. 

D ef. LXI. The Lunar Ecliptic Limit is the least distance that the moon can be at 
from one of its nodes, without being eclipsed at the time of opposition; the Solar 
Ecliptic Limit is the least distance the moon can be at from one of its nodes, without 
eclipsing the sun at the time of conjunction. 

PROP. CVIII. The solar ecliptic limit is greater than the lunar. 

Plate 10. J 3 y p r 0 p. CV. it is found, that 92' is the least latitude the moon can have, when new, without eclips- 
i lg ‘ ing the sun.* if, therefore, N c be the distance of the moon from the node, wiienuts latitude, or cC, is 
92', in the triangle C c N, the angle c CN being a right angle, because c C is perpendicular to the 
plane of the ecliptic ; the angle c NC being about 5° 30', the inclination of the moon’s orbit to the 
plane of the ecliptic; and the side cC being 92', I\l c will be found by trigonometry to be about 1 G°. 
In the same manner, supposing C c, the latitude of the moon, to be 61', according to Prop. LXXX1X. 
the length of the side N c, or the lunar ecliptic limit, will be found to be about 12 °.* Whence the 
truth of the Proposition is manifest. 

Cor. There arc more eclipses of the sun, in a course of years, than of the moon ; for the sun will 
be eclipsed, if, when the moon is new, it is within 16° of one of the-nodes, but the moon only when at 

♦The moon’s latitude may sometimes be several minutes less than that mentioned in the test, without producing an 
eclipse. 


Chap. V. 


OF THE MOON. 


193 


the full it is within 12° of one of the nodes; the sun may be eclipsed vvnile the moon is in. 64 degrees 
of its orbit, but the moon only while it is in 48 degrees of its orbit. 

Schol. Every eclipse of the moon will be visible, if the moon be above the horizonr at the time of 
the eclipse; because that part of the moon on which the shadow of the earth falls, must appear obscur¬ 
ed wherever the disk of the moon is visible. But the sun may be eclipsed, and yet the eclipse be in¬ 
visible in places to which the sun is above the horizon, because there can be no eclipse of the sun ex¬ 
cept in those parts of the earth which are within the shadow or penumbra of the moon, and neither of 
these is large enough to cover the whole disk. Hence, in any given place, more eclipses of the 
moon than of the sun will be seen in a course of years ; for though there are more eclipses of the sun 
than of the moon, many of the former are not visible at any one place while the sun is above the hori¬ 
zon ; but all the latter are visible at the same place while the moon is above the horizon.* 

PROP. CIX. When the moon is near the first of Aries, and is moving toward 
the tropic of Cancer, the time of its rising will vary- but little for several days 
together. 

If the moon were to move in the equator, its motion in its orbit, by which it describes a revolution, 
in respect of the sun, in 29 days 12 hours, would carry it every day eastward from the sun about 12° 
11 '; whence its time of rising would vary daily about 50 minutes. But, because the moon’s orbit is 
oblique to the equator, nearly coinciding with the ecliptic, different parts of it make different angles 
with the horizon, as they rise or set, those parts which rise with the smallest angles setting with the 
greatest, and the reverse. Now the less this angle is, the greater portion of the orbit rises, in the 
same time. Consequently, when the moon is in those parts which rise or set with the smallest angles, 
it rises or sets with the least difference of time, and the reverse. But in northern latitudes, the small¬ 
est angle of the ecliptic and horizon is made when Aries rises and Libra sets, and the greatest when 
Libra rises and Aries sets; and therefore, when the moon rises in Aries, it rises with the least difference 
of time. Now the moon is in opposition in or near Aries, when the sun is in or near Libra, that is, in 
the autumnal months; when, the moon rising in Aries, whilst the sun is setting in Libra, the time of its 
rising is observed to vary only two hours in 6 days in the latitude of London. This is called the harvest 
moon. 

Schol. This circumstance takes place every month; but as it does not happen at the time of full 
moon, there is no notice taken of it. When the moon’s right ascension is* equal to six signs, that is, 
when she is in or about the beginning of Libra, there is the greatest difference of the times of rising, 
viz. about an hour and 15 minutes. Those signs which rise with the least angle set with the greatest, 
and the contrary; therefore, when there is the least difference in the times of rising, there is the 
greatest in setting, and xice versa. 

The following table shows the daily mean difference of the moon’s rising and setting, on the parallel 
of London, for 28 days; in which time the moon finishes her period round the ecliptic, and gets 9 de¬ 
grees into the same sign from the beginning of which she set out. 


! = 

v: 

Signs. 

Degrees. 

Rising 

Diff. 

Setting 

Diff. 

Days. 

Signs. 

Degrees. 

Rising 

Diff. 

Setting 

Diff* 

cn 

H. 

M. 

H. 

M. 

H. 

M. 

H. 

M. 

1 

23 

13 

l 

5 

0 

50 

15 

<5 

17 

0 

46 

1 

5 

2 


26 

l 

10 

0 

43 

16 

vvy 

1 

0 

40 

1 

8 

3 

& 

10 

1 

14 

0 

37 

17 


14 

0 

35 

1 

12 

4 


23 

l 

17 

0 

32 

18 


27 

0 

30 

1 

15 

5 


6 

1 

16 

0 

28 

19 

X 

10 

0 

25 

1 

16 

6 


19 

1 

15 

0 

24 

20 


23 

0 

20 

1 

17 

7 

-A- 

2 

1 

15 

0 

20 

21 

cp 

7 

0 

17 

1 

16 

8 


15 

1 

15 

0 

18 

22 


20 

0 

17 

1 

15 

9 


28 

l 

15 

0 

17 

23 

a 

o 

O 

0 

20 

1 

15 

10 


12 

1 

15 

0 

22 

24 


16 

0 

24 

1 

15 

11 


26 

1 

14 

0 

30 

25 


29 

0 

30 

1 

14 

12 

t 

8 

1 

13 

0 

39 

26 

n 

13 

0 

40 

1 

13 

13 


21 

1 

10 

0 

47 

27 


26 

0 

56 

1 

7 

14 


4 

1 

4 

0 

56 

28 

15 

9 

1 

00 

1 

58 


* For the calculation and projection of lunar and solar eclipses, see the Problems immediately preceding the Tables at 
the end of tb\s book. 


25 





















194 


OF ASTRONOMY. 


Book VII. Part I. 


Plate 10/ 
Fig. 14. 


Plate 15. 
Fig. 1. 


Exp. Let small patches be placed on the ecliptic of a globe, as far from one another as the moon 
moves from any point of the ecliptic in 24 hours, that is, about 13*- degrees ; then, while the globe is 
turned round, observe the rising and setting of the patches in the horizon ; the hour index will show 
the difference of time at which the moon rises or sets in different parts of its orbit. 


CHAPTER VI. 

Of the Satellites o f Jupiter , Saturn, and the Ilerschel Planet. 

PROP. CX. Any satellite is at its greatest elongation from its primary, when a 
line drawn from the earth through the satellite is a tangent to the orbit of tiie satel¬ 
lite. 

Let FIE be a part of the orbit of the primary planet, AXBT the earth’s orbit, S the sun, KGNL 
the orbit of a satellite. If the earth is at X, and the satellite at L or N, so that a line XL or XN, drawn 
from the earth, is a tangent to the orbit KGNL, it may be shown, as before concerning the planets, 
that L or N is the greatest elongation of the satellite. 

PROP. CXI. Any satellite appears in inferior conjunction with its primary, when 
the satellite is between the earth and the primary, and in superior conjunction, when 
the primary is between the satellite and the earth. 

If the earth be at X, and the planet at I, the outermost satellite will be in conjunction with its 
primary when they both appear in the same line M1V, and in its inferior conjunction at M, and its 
superior at V. 

PROP. CXII. The apparent motion of any satellite, as it passes from its great¬ 
est elongation on one side of its primary through the superio conjunction, to its 
greatest elongation on the other side, is direct. 

As the satellite passes from L, its greatest elongation on one side hroug. V, its superior con¬ 
junction to N, its geocentric motion is from west to east, or in conscr u>< *■ -‘as shown concerning 

the planets. 

PROP. CXIII. The apparent motion of any satellite, . -hisses from its great¬ 
est elongation on one side of its primary through the i fcrior conjunction, to its 
greatest elongation on the other side, is retrograde. 

As the satellite passes from N through M to L, its geocentric moti u'l be f. m east to west, 
or in antecedentia, as was proved concerning the planets. 

Cor. The satellites are sometimes to the west and sometimes to the et r ’ (heir primaries. 

PROP. CXIV. The greatest elongations of a satellite on each . kie tire equal. 

For by observation it is found, that the angles LXI, NX1, are equal with respect u the satellites. 

Cor. Hence it appears that the orbits ol the satellites are circular, or ne \ h,t ing their pri¬ 
maries at the centre of their orbits. 

Schol. The satellites move round the primary in orbits, that are nearh c . dav but round the 
sun in curves of a different kind. Let ABCDE, &c. to T, (Plate 15, Fig. 1.) - « ch of Jupiter’s 
orbit as is described by that planet in 18 days. Then the curves a, b , c, d , (accoi Ferguson) 

represent the paths of the four satellites. 

If we suppose Jupiter to move from A, the first satellite from a , the sec< the third 

from c, and the fourth from d; or each of the satellites from the point of conjun he sun, as 

seen from the primary; then, at the end of one day, Jupiter will be at B, and ht lib s at 1, in 
their respective courses. At the end of the second day Jupiter will be at C, ; ‘ ' ites at 2 

in the curves they respectively describe, and so on ; the capital letters showing i :e in its 



Chap. VI. 


OF THE SATELLITES. 


path at the end of each day, the figure under each of them the number of the day, and the like figures 
on the paths of the satellites, their places at the same time. The first satellite appears to be station¬ 
ary at -j- near C, as seen from the sun; retrograde from + to 2 , at 2 stationary again; thence direct till 
beyond 3; twice stationary and once retrograde between 3 and 4. This satellite intersects its own 
path every 42i hours, making loops as in the diagram at 2, 3, 5, &c. soon after every conjunction. 
The second crosses its own path every 3 days 13 hours, as at 4, 7, 11, &c. making only 5 loops, and 
as many conjunctions while the first makes 10 . The third satellite at the end of every 7 days 4 hours, 
makes an angle in conjunction with the sun, as at 7, 14. The fourth satellite is always progressive, 
making neither loops nor angles; its first conjunction is at e, at the end of 16 days 18 hours. The 
first, second, and third are, according to the figure, nearly in the same relative situation, every 
seventh day. 

The path of Saturn’s first satellite about the sun is looped, but not those of the second, third, fourth, 
and fifth. 


PROP. CXV. The satellites of Jupiter, Saturn, and the Herschel, are eclipsed by 
their respective primaries. 

The planet I, being an opaque body, casts a shadow IV, opposite to the sun. Therefore, when one PJ ate 
of its satellites in describing the arc VO comes to V, it will be eclipsed by falling into this shadow. If * ‘ s 
the earth is at A in its orbit, a spectator from the earth will lose sight of the satellite, when it is thus 
eclipsed at V ; and then, as it emerges from the shadow, it becomes again visible, till, at O, it passes 
behind its primary. If the earth be at X, the satellite will be eclipsed, and in occultation at the same 
time. 


PROP. CXVI. When one of the satellites passes between the sun and its primary, 
it eclipses the sun. 

A satellite at M will be between the sun and its primary, and occasion an eclipse of the sun on that 
part of the primary where the shadow of the satellite passes, which shadow will appear as a dark spot 
on the disk of the planet to an inhabitant of the earth. 

Schol. Dr. Herschel has discovered, that the fifth satellite of Saturn is, in its rotation, subject to the 
same law that our moon obeys, that is, it turns round its axis in the same time in which it revolves 
about the planet. Hence the doctor thought it natural to conclude, that all the secondary planets might 
be governed by the same laws to which those are subject. This theory he thinks considerably confirm¬ 
ed by certain observations, which he made on the satellites of Jupiter, and which he communicated to 
the Royal Society, June 1 , 1797. 

The following table will give the periodical times and distances of Jupiter’s satellites, and the angles 
under which their orbits are seen from the earth, as its mean distance from Jupiter. 


Satellites. 

Days. 

h. 

min. 

Dist. in miles. 

Angles 

of orbit. 

1 — 

1 

18 

27.6 • 

— 266,000 

— 3' 

55" 

2 — 

3 

13 

13.7 ■ 

— 423,000 

— 6 

14 

3 — 

7 

3 

42.6 - 

— 676,000 

— 9 

58 

4 — 

16 

16 

31.8 - 

— 1.189,000 

— 17 

30 


These satellites of Jupiter are of great use in astronomy. (1.) In determining the distance of Jupiter 
from the earth. (2.) They afford a method of demonstrating that the motion of light is progressive, and 
not instantaneous, as was once supposed. See Prop. CXVII. And (3.) The most considerable advantage 
is derived from the eclipses of the satellites of Jupiter, in ascertaining the longitude of different places 
on the earth. 

Exp. Suppose two observers of an eclipse, the one at London, the other at the Cape of Good Hope ; 
the eclipse will appear to both at the same moment of time ; but being situated under different me¬ 
ridians, they count different hours, according to which their difference of longitude is found. Thus, if 
at the Cape of Good Hope, an emersion of a satellite is observed at lOh. 46' 45" apparent time, and the 
same is seen at Greenwich at 9h. 33' 12'', the difference of which times is lh. 13' 33'^, the longitude of 
the cape east of Greenwich in time, or 18° 23' 15". 

Note. The third satellite is the largest of all; the first and fourth are nearly of the same size ; the 
second is the smallest. 


196 


OF ASTRONOMY. 


Book VII. Part I. 


OF THE SATELLITES OF SATURN. 


Satellites, 

Periodical times. 


Distance in miles. 

i — 

1 day. 

2 lh. 

18' 

27" — 

170,000 

2 — 

2 

17 

41 

22 — 

217,000 

3 — 

4 

12 

25 

12 — 

303,000 

4 — 

15 

22 

41 

13 — 

704,000 

5 — 

79 

7 

48 

— —: 

2.050,000 

6 — 

1 

8 

53 

8 — 

135,000 

7 — 

— 

22 

37 

22 — 

107,000 


Plate 14. Fig. 3. Plate 14, is a view of the proportional magnitudes of the orbits of the moon, Jupiter’s four 
Fig. 3. satellites, and the tive first of Saturn. Mm represents the moon’s orbit, the earth being supposed to 
be at E; J Jupiter, 1, 2, 3, 4, the orbits of the four satellites; Sat. Saturn, 1 , 2, 3, 4, 5, the orbits of 
the five first satellites. 

The 6 th and 7th satellites were discovered by Dr. Herschel, in the years 1787 and 1788. To pre¬ 
vent mistakes, he calls them the 6 th and 7fh, though nearer to the planet than the other five. Dr. 
Herschel observes, that Saturn has probably a considerable atmosphere. It turns on an axis perpen¬ 
dicular to the ring, in 10 h. 16' 0.44 ', and is flattened at the poles, so that the equatorial diameter is 
to the polar as 11 to 10. Phil. Trans. Vol. 80, Part 1. and II. and Vol. 84. 

OF THE SATELLITES OF HERSCHEL. 

Dr. Herschel has at different times discovered six satellites belonging to the new planet, two of 
which he described in the 77th and 78th Vols. of the Phil. Trans, where he says, “I confess that this 
scene appeared to me with additional beauty, as the little secondary planets seemed to give a dignity 
to the primary one, which raises it into a more conspicuous situation among the great bodies in our 
system.” 

The following is the arrangement of the six satellites. 

Satellites, When discovered. Periodical times. 


1 

— 

Jan. 

18, 1790 

— 5d. 

21 h. 

25' 

0 " 

2 

— 

Jan. 

11, 1787 

— 8 

17 

1 

19 

3 

— 

Mar. 

26, 1794 

— 10 

23 

4 

0 

4 

— 

Jan. 

11, 1787 

— 13 

11 

5 

\\ 

5 

— 

Feb. 

9, 1790 

— 38 

1 

49 

0 

6 

— 

Feb. 

28, 1794 

— 107 

16 

40 

0 


*“It will be hardly necessary,” says Dr. Herschel, ‘ c to add, that the accuracy of these periods de¬ 
pends entirely upon the truth of the assumed distances ; some considerable difference, therefore, may be 
expected, when observations shall furnish us with proper data for more accurate determinations.” 
See Phil. Trans. 1798. 

PROP. CXVII. A ray of light is about 8 minutes in coming from the sun to the 
earth. 

Plate 10 . Let Abe the sun, BECD the earth’s orbit, F the planet Jupiter, and PING the orbit of its inner 
Fig. 13. satellite. Let FGH represent the shadow of Jupiter. While the satellite is between IT and G it is 
eclipsed; when it comes to IT, it emerges, and becomes visible to a spectator at B. From comparing 
the times of the apparent entrance and emersion of the satellite, with tables calculated for the mean 
distances of the earth from the satellite, the visible emersion at the least distance is found to happen 
about 8 minutes sooner, and at the greatest distance about 8 minutes later, than by the tables; conse¬ 
quently, the ray of light is about 16 minutes in passing through the earth’s orbit, or 8 minutes in com¬ 
ing from the sun to the earth. 

Cor. The diameter of the earth’s orbit being 190,000,000 miles, the velocity of light will be 
190,000,000 

——-—— = 197,916| miles in a second. 

16x60 ’ 3 

PROP. CXVIII. Jupiter is surrounded by cloudy substances, subject to frequent 
changes in their situation and appearance, called his Belts. Saturn is encompassed 
with a Ring, whose greatest apparent diameter is to that of the planet as 9 to 4. 

Plate 10 . These are known from observation. The Belts of Jupiter are sometimes of a regular form; some- 
Fig. 15 . times interrupted and broken; and sometimes not at all to be seen. The plane of Saturn’s ring is 



Chap. VII. 


OF COMETS. 


197 


inclined to the plane of the ecliptic at an angle of 31 degrees; which appears like two arms to the 
planet, and is only visible when the sun and the earth are both on the same side of its plane. On account 
of its inclination, it always appears oblique to the eye, and therefore elliptical; whence the part 
behind Saturn is invisible, and the part before cannot be distinguished from the planet. The ring, 
being opaque, can only be visible when the sun’s rays are reflected from its broad surface to the earth, 
that is, when the sun and the earth are both on the same side of the plane of the ring. 

The latter discoveries of Dr. Herschel have shown, that what was supposed to be a single broad 
flat ring of Saturn, is divided into two parts, lying exactly in the same plane, and revolving about 
an axis perpendicular to that plane, in 10 h. 32' 15''. The dimensions of these concentric rings, and 
the space between them, he states to be as in the following table. 


Miles. 

Inner diameter of the smaller ring - - - 140,345 

Outside diameter of ditto - - - - 184,393 

Inner diameter of the larger ring - - - - 190,248 

Outside diameter of ditto ------ 204,883 

Breadth of the inner ring ------ 20,000 

Breadth of the outer ring - - - - - - 7,200 

Breadth of the vacant space ------ 2,830 


CHAPTER VII. 

Of Comets . 

PROP. CXIX. Comets are opaque and solid bodies. 

A comet, at a given distance from the earth, shines much brighter when it is on the same side of 
the earth with the sun, than when it is on the contrary side ; from whence it appears that it owes its 
brightness to the sun. The resistance of the comet of 1680 to the action of the great heat, to which 
it was probably exposed in its near approach to the sun, furnishes evidence in favour of its being a 
fixed and solid body. 

PROP. CXX. The comets describe very eccentric ellipses about the sun, placed 
in one of their foci. 

They are observed to approach toward, and afterward recede from, the sun, and to describe paths 
in the heavens, which agree with elliptic orbits; it is therefore most probable, that, agreeably to 
the general analogy of nature, they move in such orbits, and have the sun in one of the foci of the 
ellipse. The calculations framed upon this supposition, by which the returns of comets have 
been foretold, having, as far as observations have been made, been found to agree with the phenom¬ 
ena, strongly confirm the truth of the Proposition. 

Schol 1. Comets are often accompanied with a luminous train, called the tail, which is conjec¬ 
tured to be smoke rising from the body in a line opposite to the sun. The body of the comet is 
supposed to he surrounded by an atmosphere ; the sun is also supposed to be surrounded by an ether, 
or a subtle fluid, extending to a great distance from the sun, which may be considered as the solar 
atmosphere. From the heat which the comet has acquired by approaching toward the sun, and by 
the reflection of the sun’s rays from the solid body and atmosphere of the comet, the parts of the 
solar atmosphere where the comet passes are more heated, and consequently more rarefied or spe¬ 
cifically lighter than elsewhere. The paris thus rarefied will be put into motion; and since there will 
be a constant succession of fresh portions of the sun’s atmosphere within that of the comet, there will 
be a perpetual stream of this rarefied matter. This stream will impel the particles of the comet’s 
atmosphere, and make them move along with it, thus producing the smoke which, reflecting the sun’s 
rays, forms the visible tail. And this stream of rarefied solar atmosphere will move those parts of this 
atmosphere which have the least specific gravity, that is, directly from the sun. 

Schol. 2. Of all the comets, the periods of only three are known with any degree of certainty. The 
first of these comets appeared in the years 1531, 1607, and 1682; and is expected to appear every 
75 th year. The second of them appeared in 1532 and 1661, and was expected to return in 1789, and 
every 129th year afterward. The third, having last appeared in 1680, and its period being no less 





198 


OF ASTRONOMY. 


Book VII. Part. I. 


than 575 years, cannot return until the year 2225. This comet, at its greatest distance, is about 1! 
thousand two hundred millions of miles from the sun; and at its least distance from the sun’s centre, 
which is 49,000 miles, is within less than a third part of the sun’s semidiameter from his surface. In 
tliat part of its orbit which is nearest the sun, it moves at the rate of 880,000 miles in an hour. 

Schol. 3. Dr. Halley, who saw the comet which appeared in 1682, observes, u that there are many 
things which make me believe, that the comet which Apian saw in the year 1531, was the same with 
that which Kepler and Longomontanus more accurately described in the year 1607 ; and which I my¬ 
self have seen return, and observed in the year 1682. All the elements agree, and nothing seems to 
contradict this opinion besides the inequality of the periodic revolutions; which inequality is not so 
great, but that it may be owing to physical causes. For the motion of Saturn is so disturbed by the 
rest of the planets, especially Jupiter, that the periodic time of that planet is uncertain for some whole 
days together. How much more, therefore, will a comet be subject to such like errors which rises al¬ 
most four times higher than Saturn, and whose velocity, though increased but very little, would be suf- 
ficent to change its orbit from an ellipse to a parabola. And I am the more confirmed in my opinion 
of its being the same; for, in the year 1456, in the summer time, a comet was seen passing retrograde 
between the earth and sun, much after the same manner; which, though nobody made observations 
upon if, yet, from its period, and manner of transit, I cannot think different from those I have just now 
mentioned. And since looking over the history of comets, I find, at an equal interval of time, a comet 
to have been seen about Easter in the year 1305, which is another double period of 151 years before 
the former. Hence I think I may venture to fortell that it will return again in the year 1758.” 

Dr. Halley computed the effect of Jupiter upon this comet in 1682, and found that it would in¬ 
crease its periodic time above a 3 r ear; in consequence of which, he predicted its return at the end of 
the year 1758, or the beginning of 1759. M. Clairaut computed the effects of both Saturn and Jupiter, 
and found that the former would retard its return in the last period 100 days, and the latter 511 days ; 
and he determined the time w r hen the comet would come to its perihelion to be on April 15, 1759; 
observing, that he might err a month from neglecting small quantities in the computation. The comet 
did pass the perihelion on March 13, within 33 days of the time computed. Now, if Dr. Halley meant 
the time of its passing the perihelion, and we add 100 days for the action of Saturn, which he did not 
consider, it will bring it very near to the time in which it passed the perihelion, and prove his compu- 
taion of the effect of Jupiter to have been very accurate. But if he meant the time when the comet 
would first appear, his prediction was accurate, for it was seen on December 14, 1758. Dr. Hallev, 
therefore, had the glory first to fortell the return of a comet, and the event answ ered, in a remarkable 
manner, his prediction. He farther observed, that the action of Jupiter, in the descent of the comet 
toward its perihelion in 1682, would tend to increase the inclination of its orbit; and accordingly the 
inclination in 1682 was found to be 22' greater than in 1607. 

Dr. Halley suspected the comet in 1680, to have been the same which appeared in 1106, 531 and 
44 years before Christ. He also conjectured, that the comet observed by Apian, in 1532, was the same 
as that observed by Hevelius in 1661 ; if so, its period was 129 years, and it ought to have returned in 
1789, but it did not appear. M. Mechain having collected all the observations in 1532, and calculated 
the orbit again, found that it differed materially from that calculated by Dr. Halley, which renders it 
extremely doubtful whether this was the comet which appeared in 1661 ; and this doubt is increased by 
its not appealing in 1789. 

Schol. 4. From the beginning of our era to this time, it is probable, according to the best accounts 
that there have appeared about 500 comets. Before that time above 100 others are recorded to have 
been seen, but it is probable that not above half of them were comets. And when we consider that 
many others may not have been perceived, from being too near the sun ;—from appearing in moonlio-ht- 
—from being in the other hemisphere;—from being too small to be perceived ; or which may not have 
been recorded, we might imagine the w'hole number to be considerably greater; it is, however, highly 
probable, that of the comets which are recorded to have been seen, the same may have appeared sever¬ 
al times, and therefore the number may be less than is stated. Miss Caroline Herschel, the sister of Dr 
Herschel, has discovered several comets within the last 15 years, accounts of which are in the differ¬ 
ent volumes of the Philosophical Transactions. 

On the subject of Comets, see Mr. Vince’s very excellent “ Complete System of Astronomy ,” Vol. I 
Quarto. 


Chap. VIII. 


OF THE SUN. 


199 


CHAPTER VIII. 

Of the Sun. 

PROP. CXXI. The spots, which appear upon the sun’s disk, adhere to its surface. 

If one of these spots appears upon the eastern limb or edge of the sun’s disk, it moves from thence 
toward the western edge, and arrives at the western edge in about 13^ days. Here the spot disappears; 
and in about 13§ days more, it is seen again upon the eastern edge ; and so continues to go round, com¬ 
pleting its apparent revolution in 27 days; during one half of which time we see it on the disk of the 
sun, and during the other half it disappears; which could not happen, if the spots did. not adhere to the 
surface of the sun. Let A be the centre of the sun’s disk, D its eastern, and C its western edge ; HEG 
the orbit of an opaque body moving round it, and B the eye of the spectator at the earth. If two lines 
BD and BC are supposed to be drawn from the spectator’s eye B, so as to touch the sun at D and C, 
then DBC, the angle contained between these lines, is the angle under which the sun’s diameter appears 
to a spectator on the earth. EG is the only part of the supposed body’s orbit that is within this angle 
DBC; and consequently, if the body was in any other part of its orbit, except EG, it would not appear 
upon the sun’s disk. But EG is less than half its orbit; and the body would not take up half the time 
of a revolution to describe EG. Therefore such a body would not be seen upon the sun’s disk, as the 
spots are, for half the time of a revolution. But if the orbit HEG is not greater than LDFC, or is close 
to the sun ; that is, if the spot adheres to the sun’s surface, then half its orbit DEC will be within the 
angle DBC, and, consequently, the spot will appear upon the sun’s disk during one half of its revolu¬ 
tion; but during the other half of its revolution, while it describes CLD, it will disappear, because then 
it will be behind the sun, and so will be concealed from the earth ; which agrees with the phenomena. 

PROP. CXXII. The sun is a spherical body, which revolves upon its axis from 
west to east. * 

The spots, which appear in the sun’s disk, adhere to its surface, (by Prop. CXXI.) and those spots 
xevolve ; therefore the sun revolves round its axis. 

Whatever side of the sun is turned toward the earth in this rotation, it always appears to be a flat, 
bright circle ; but all the sides of it could not appear in this manner unless it was a sphere; therefore 
the sun is a spherical body. fMj 

Schol. A real revolution of a spot, and consequently of the sun round its axis, is completed in 25 
days, two days less than its apparent revolution, in consequence of the earth’s motion in its orbit in the 
same direction in which the spot moves. 

PROP. CXXIII. The axis of the sun is inclined to the plane of the ecliptic. 

Each spot upon the sun must describe a circle round the sun, either coinciding with its equator, or 
parallel to it. If therefore the sun’s axis were perpendicular to the plane of the ecliptic, the plane of 
the sun’s equator would be in the plane of the ecliptic; and a spectator on the earth, whose eye is in 
the ecliptic, would see the spots describing right lines, either in the sun’s equator, or parallel to it; 
but the spots are sometimes seen to describe lines oblique to the plane of the ecliptic; therefore the 
axis of the sun is inclined to the plane of the ecliptic. This inclination is observed to be an angle of 
about 82i degrees. When the sun is in that part of the ecliptic where its equator crosses the plane of 
the ecliptic, the spots appear to describe right lines parallel to the sun’s equator. 

Schol. The following particulars respecting the sun are related by Sir I. Newton. 

1 . That the density of the sun’s heat, which is proportional to his light, is 7 times as great in Mer¬ 
cury as with us, and that water there would be all carried off in the shape of steam ; for he found by 
experiments with the thermometer, that a heat 7 times greater than that of the sun’s beams in summer 
will serve to make water boil. 

2 . That the quantity of matter in (he sun is to that in Jupiter nearly as 1100 to 1, and that the dis¬ 
tance of that planet from the sun is in the same ratio to the sun’s semidiameter; consequently, that the 
centre of gravity of the sun and Jupiter is nearly in the superficies of the sun. 

3 . That the quantity of matter in the sun is to that in Saturn as 2360 to 1, and that the distance of 
Saturn from the sun is in a ratio but little less, than that of the sun’s semidiameter. And hence the 
common centre of gravity of Saturn and the sun is a little within the sun. 

4 . By the same method of calculation it will be found, that the common centre of gravity of all the 
planets cannot be more than the length of the solar diameter distant from the centre of the sun. 

5. The sun’s diameter is equal to 100 diameters of the earth, and therefore its magnitude must 
exceed that of the earth one million of times. 


Plate 10. 
Fig. 16. 


200 


OF ASTRONOMY. 


Book. VII. Part I. 


6 . If 360 degrees (the whole ecliptic) be divided by the quantity of the solar year, it will give 59' 
8 ", which therefore is the medium quantity of the sun’s apparent daily motion ; hence his horary motion 
is equal to 2' 27". By this method the tables of the sun’s mean motion are constructed, as found in 
astronomical books. 


CHAP. IX. 

Of the Parallaxes, Distances , and Magnitudes of the Heavenly Bodies. 

D ef. LXII. The Parallax of the heavenly bodies is the change of their apparent 
situation with respect to each other, as the spectator views them from different 
stations. 

Def. LXIII. The Diurnal Parallax is the distance between the apparent place of 
a heavenly body, as viewed from the surface of the earth, and its apparent place, as 
viewed from the centre of the earth. 

’late 10 . Let DAB be the earth, C its centre, A the station of a spectator on the surface of the earth; and 
ig- 17- F, G, H, ditferent places of the moon, or any other heavenly body; TO, NM, LI, are its different par¬ 
allaxes, and THO, or AHC ; MGN", or AGC, &,c. angles of parallax. 

Schol. If a spectator in his first station at A, sees a planet at G, its apparent place in the heavens 
will be N ; if now, by the diurnal rotation of the earth, he comes into the station P, the planet will 
appear at M, which is the place in which it would have appeared if viewed from C the centre ; thus, 
in all cases, the parallax which arises from the diurnal motion, is the same which would arise from a 
change of station from the surface to the centre ; for in either case, the change of the spectator’s 
line of view is the same. Hence appears the propriety of the above definition of the diurnal 
parallax. 

PROP. CXXIV. The parallax of any planet is always proportional to the angle 
which a semidiameter of the earth, drawn from the station of the spectator upbn the 
surface to the c^ptre, would subtend, if viewed from the planet. 

Plate 10 . jf t j, e pl ane t be at H, and the spectator at A, AHT will be his line of view; on changing the sta- 
17 ’ lion of the spectator from A to C, the line of view* will become CHO ; whence TO will be the paral¬ 
lax. But TO subtends and is proportional to THO, or (El. I. 15.) AHC, the angle which the earth’s 
semidiameter would subtend, if viewed from the planet H. 

PROP. CXXV. The parallax of a planet depresses its apparent place, by the par¬ 
allactic arc. 

Plate 10 . If the planet be viewed from C, its apparent place is O ; if from A, its apparent place is T, farther 
Fig. 17. from Z the vertex than O, by the parallactic arc TO. 

Cor. When the altitude of a body is observed, it must be corrected by parallax and refraction, 
adding the former, and subtracting the latter, in order to get the true altitude, or the altitude above the 
rational horizon at the centre of the earth. Dee the table of refractions, Schol. 1 Prop. XXXIX. 

PROP. CXXVI. The diurnal parallax of any planet, at a given distance from 
the earth, is greatest when the planet is in the horizon, and decreases as the altitude 
of the planet increases. 

The parallax (by Prop. CXXIV.) is proportional to the angle which AC would subtend, if seen from 
the planet H ; but this given line, viewed from the given distance of the planet, would continually di¬ 
minish in its apparent magnitude (by Book VI. Prop. LXXI1I.) as the degree of obliquity at which it is 
viewed increases; that is, as the planet advances from H toward E ; therefore the parallax is greatest in 
the horizon, and decreases as the planet approaches the vertex. The parallactic angle AGC is less 
than AHC, and AFC less than AGC. 

PROP. CXXVII. To find the parallax of the moon, or any planet. 

Plate 10 . Let HMO be an arc of the horizon ; APVM an arc in the meridian; P the elevated pole ; V the 
Fig. 18. 



Chap. IX. 


201 


OF PARALLAXES, &c. 

vertex; E the apparent place of the planet, as seen from the surface of the earth, and S its place, as 
seen from the centre ; then ES is the diurnal parallax in the vertical circle VE. Before the planet 
comes to the meridian, observe its altitude, at E, above the horizon, whence the complement of its 
altitude, VE, will be known; at the same time observe its distance from the meridian, or its azimuth, 

EVM. After the planet has passed the meridian, observe when it has the same altitude as at the first 
observation; that is, when e V is equal to EV. Now, if E is the apparent place of the planet when 
at the time of the first observation it is viewed from the earth’s surface, and S would be its place, at 
that time, if viewed from the centre; and if e is its apparent place when viewed at the second observa¬ 
tion, from the surface, and s would be its place, at that time, if viewed from the centre ; the parallax 
ES is equal to the parallax es, since the altitude was the same at both observations, and consequently 
SV is equal to s V. So that if PS is the secondary of the equator which passed through the planet at 
the first observation, and P s the secondary which passed through it at the second observation, the planet 
between the times of the first and second observation, must have described the arc S s in a circle of daily 
motion. From the time which has passed between the two observations, the arcSs (by Prop. XXVII.) 
may be found, and consequently the angle SPs. Now, because the angle EVM is known, PVS, its sup¬ 
plement to two right angles, is known ; and, because at the two observations the planet was at equal 
altitudes, that is, at equal distances from the meridian, the meridian bisects the angle SP s, which is 
known; whence its half VPS is found.* Also, if the latitude of the place be known, PV, the distance of 
the elevated pole from the vertex, or the complement of its distance from the horizon, that is, (by Prop. 

III.) the complement of latitude, is known. Therefore in the spherical triangle PVS, two angles and 
one side are kftown; whence the length of SV may be determined. Take SV from EV, which is already 
known, and SE, the planet’s parallax, will be found. The moon’s mean parallax has been found to 
be 57' 11". 

Or thus ; observe when the planet, whose parallax is to be found, and any fixed star in conjunction 
with it, cross the meridian at the same instant; observe the same planet and star after three hours, and 
remark how much sooner the planet reaches a line placed perpendicularly in the telescope than the star. 

As 24 hours is to this difference of time, so will 360 degrees be to the arc which subtends the angle of 
the parallax; whence the arc and angle will be known. 

PROP. CXXVIII. Any parallax of a planet being given, to find any other parallax. 

The paraliactic angle AFC being given, it is required to find the angle AHC. Having measured Plate 10, 
the angle ZAL, let the angle ZAH, the apparent distance of the planet from the zenith, be also meas- 17 ‘ 
ured. Then, in the triangle CAF, the sine of the angle CAF is to the sine of the angle CPA, as the 
side CF is to the side AC. Again, in the triangle CAH, the sine of the angle CAH is to the sine of 
the angle CHA, as CH is to AC. But CH is equal to CF; therefore the sine of the angle CAF, is to 
the sine of the angle CFA, as the sine of the angle CAH is to the sine of the angle CHA; but the three 
first terms are known, therefore the fourth, namely, the angle CHA, may be found. 

PROP. CXXIX. At a given altitude of different planets, their diurnal parallaxes 
are inversely as their distances from the centre of the earth. 

Let one planet be at / where its altitude is/Ap, and another at /s, having an equal altitude h A p. Plate 10. 
If the planet/ is viewed from A at the surface of the earth, the line of view is A/ r. and r is its appar- Fl »‘ ' 
ent place in the heavens; viewed from C, its apparent place would be t; whence, its parallax (by Prop. 

CXXV.) is rt. In the same manner it may be shown, that r s, which is less than ri, is the* parallax of 
the planet h. But (by Prop. CXX1V.) the parallax of each planet is proportional to the "angle which 
AC would subtend, if viewed from the planet; and since AC is given, and also the degree of obliquity 
at which it is viewed, the. apparent length of AC, or the angle which AC would subtend, at either 
planet, would be (by Book VI. Prop. LXIX.) inversely as the planet’s distance from C. Therefore the 
parallaxes of these planets are inversely as their distances from the centre of the earth. 

PROP. CXXX. The diurnal parallax of a planet in a vertical circle generally pro¬ 
duces a parallax of declination, and also, if the planet is not in the meridian, of right 
ascension. 

Let HQ, be the horizon ; EC an arc of the equator which cuts the horizon at C ; P the pole of the Plate 11 . 
equator; Z the zenith; ZV a vertical circle; Fthe apparent place of a planet in the vertical circle ZV, F, S- s ‘ 

* The moon’s declination sometimes varies 12' or 15' an hour, which would render the consecutive distances from the 

J an at equal altitudes materially unequal, especially iu high latitudes, and thus render this method of finding the 
•noon's parallax totally useless. 

26 


202 


OF ASTRONOMY. 


Book VII. Part I. 


Dale 11. 

Fig. J. 


Plate 10. 
Fig. 17. 


Plate 10. 
Fig. 17. 


Plate 9 
Fig. 12. 


as viewed from the surface of the earth, and I its apparent place, as viewed from the centre ; then (by 
Def. LX11I) FI is the diurnal parallax in a vertical circle. When the apparent place is F, FFA is a sec¬ 
ondary of the equator passing through it, and when it is 1, FIB is the secondary which passes through it. 
Therefore AF is the declination of the planet when it appears at F, and BI its declination when it appears 
at I; the difference of which, Dl,is the change of the apparent declination arising from the different sta¬ 
tion of the spectator at the surface or centre of the earth. When the apparent place is F, the distance 
of A from the first of Aries is the right ascension ; when it is I, the distance of B from the first of Aries is 
the right ascension; for PFA and FIB are secondaries of the equator passing through the planet. The 
difference of right ascension, therefore, produced by the parallax FI is AB. If the planet is in the me¬ 
ridian PZH, and if L be its apparent place, as viewed from the surface, and N, as viewed from the centre 
of the earth, LN will be its diurnal parallax; LE its declination, as viewed from the surface ; NE its 
declination as viewed from the centre ; and NL its parallax of declination. But, because PZH is a sec¬ 
ondary of the equator, in whatever part of this vertical circle the planet appears, its right ascension will 
he the distance of the point E from the first of Aries; that is, the diurnal parallax, in this case, makes 
no parallax of right ascension. 

PROP. CXXXI. The diurnal parallax of a planet in a vertical circle generally 
produces a parallax of latitude, and also, if the vertical circle be not a secondary of 
the ecliptic, of longitude. 

Let HQ be the horizon ; P the pole of the ecliptic; EC an arc of the ecliptic, which cuts the ho¬ 
rizon at C ; and ZV a vertical circle ; and this Proposition may be proved in the same manner as the 
last. 

PROP. CXXXII. The semidiameter of the earth is to the distance of any planet 
from the centre of the earth, as the sine of the planet’s parallax is to the sine of its 
apparent distance from the vertex. 

If a planet is at F, and the spectator at A, where the line of view is AFL, the planet will appear at 
L, and ZAL will be the angle of its apparent distance from the vertex Z. Let the parallax IL, or the 
angle AFC proportional (by Prop. CXXIV.) to IL, be found. In the plane triangle ACF, (the sides be¬ 
ing to one another as the sines of the opposite angles) AC the semidiameter of the earth, is to FC, the 
distance of the planet from the centre of the earth, as the sine of the angle AFC, the angle of the par¬ 
allax, is to the sine of the angle FAC, or of its supplement to two right angles ZAL, the angle of the 
planet’s apparent distance from the vertex. 

Cor. When the horizontal parallax is taken, the semidiameter of the earth AC, is to HC, the dis¬ 
tance of the planet, as the sine of the horizontal parallax AHC is to the sine of II AC or radius. 

PROP. CXXXIII. To measure the distance of the moon from the earth. 

Let H be the moon in the sensible horizon observed by a spectator at A, and C the centre of the 
earth. In the triangle AHC, let the angle APIC, the moon’s horizontal parallax, be found, (by Prop. 
CXXV1I.) The angle HAC is a right angle, and AC, the semidiameter of the earth, is known to 
be 3956 miles. Hence AC, the sine of AHC, 57' 11", is to 3985, as AH, taken as radius, to the 
number of miles in AH, the moon’s distance from the earth ; the moon’s mean distance is thus found 
to he 238,20’0 English miles. 

Schol. According to Mr. de la Lande, the horizontal semidiameter of the moon is to its horizon¬ 
tal parallax for the mean radius of the earth, as 15' is to 54' 57'.4, or very nearly as 3 to 11 ; hence 
the semidiameter of the moon is of the radius of the earth. And as the magnitudes of spherical 
bodies are as the cubes of their radii, the magnitude of the moon is to that of the earth, as 3 3 to ll 3 , 
that is, as 1 :49. 

PROP. CXXXIV. To determine the relative distances of the inferior planets 
from the sun. 

Let S be the sun, EHG the orbit of Venus, and LCM the orbit of Mercury. Let AXF be a tan¬ 
gent to the orbit of Venus, and let the elongation of Venus, that is, the angle XAS, be found by 
observation. Then as radius to the sine of the angle XAS, so is AS to XS or ES. In like manner, 
if the elongation of Mercury, or the angle CAS, be observed; as radius to the sine of CAS, so is 


Chap. IX. 


203 


OF PARALLAXES, &c. 

AS to CS or LS. If AS the sun’s distance from the earth, he supposed to be divided into 1000 equal 
parts, the distance of Mercury wiil in this manner be found to be 387, and that of Venus 723, 

PROP. CXXXV. To determine the relative distances of the superior planets 

from the sun. 

• 

Let S be the sun, nkg the orbit of the earth, OPQ the orbit of Mars, and NKG a part of a great p| it te 10. 
circle in the heavens, in which the planet appears to have a retrograde motion ; let P he the place Fig. 1. 
of Mars. Whilst the eanh is passing in its orbit from k to n, Mars wiil appear to move from k to N. 

The angle of retrogradation KPN is then known by observation. To this the vertical angle n PS is 
equal. In the triangle n SP, the angle at n is a right angle; the angle n PS is the angle of retrogra- 
dation which is known, whence the other angle n SP is known, and the ratio of the sides of the trian¬ 
gle to each other is known ; whence the ratio of S n to SP is found. Ifthe mean distance of the earth from 
the sun be called 1000, that of Mars will be found to be 1523, that of Jupiter 5203, and that of Saturn 
9539. 

PROP. CXXXVI. To find the parallax of the sun by the transit of Venus. 

In order to explain the general principles of this operation, it must he understood, that the peri- p ! ate h. 
odical times of Venus and the earth, and the proportions and positions of their orbits, are accurately * 
known from previous observations. From these elements are computed the horary motion o*' both 
planets, the latitude of Venus, the direction and length of its path across the sun’s disk, and the dura¬ 
tion of the transit as viewed from the earth’s centre. Then let S be the sun, B'E'e a part of the 
orbit of Venus, its apparent motion being retrograde, or from left to right, (see Prop. LIX.) and OVV 
a part of the earth’s orbit; and let the transit be observed from some place on the earth’s surface, 
where the sun, for the greater advantage, is on the meridian, about the middle of the transit; and, the 
earth being at O, let b represent the situation of that place at the beginning of the transit, when Venus 
at B' is seen just entering on the sun’s disk at B. If that place should continue stationary with res¬ 
pect to the earth’s centre, Venus must reach e' in its orbit at the end of the transit when it is appar¬ 
ently passing off from the sun’s disk at E, but during this time the place is carried from b to the 
eastward by the earth’s diurnal motion to the situation e, where it is at the end of the transit, so that 
the planet passes off from the sun’s disk when it has only reached E' in its orbit, the duration of the 
transit, as computed for the earth's centre, being shortened by the motion of the place of observation 
in the ratio of B' e to B'E'. Now as one hour is to this difference between the computed and observ¬ 
ed durations, so is the heliocentric horary motion of Venus from the earth (or the difference between 
the heliocentric horary motions of Venus and the earth) to th» arc E'e' or the angle eEb subtended 
at the sun by the line eb. This line may be computed from the latitude of the place of observation 
and the observed duration of the transit, and by comparing it with the semidiameter of the earth, 
allowing for the obliquity of the angles at b and e, we may obtain the angle subtended at the sun by 
the semidiameter of the earth, or the sun’s horizontal parallax Thus an angle so small as to be 
scarcely measurable, if it were directly accessible, is determined with great accuracy by observing 
the time in which an arc, subtending it, is described by a motion so slow, as to afford an interval of 
time amply sufficient for determination. In fact, it is found that if the line cb should be equal to the 
earth’s semidiameter, and perpendicular to e E, Venus, with only the excess of its heliocentric horary 
motion above that of the earth, would be more thah five minutes in passing over the arc E' e' subtend¬ 
ing the angle b Ec, which would then be exactly equal to the sun’s horizontal parallax. 

But lest there should be some error in the computed length of the transit, it ought to be observed 
at some place near the meridian opposite to the former, and at such a distance from the enlightened 
pole that the beginning may be observed before sunset and the end after sunrise, so that during the 
transit the observer may be carried in a direction, with respect to the sun and planet, opposite to 
that in which the former was carried. Let W now denote the earth during the transit, and b the 
place of the observer at the beginning when the planet at B' is apparently just e tering on the sun’s 
disk at B; if the observer, as before, should continue at 6, Venus would perform the transit, describing 
the computed chord of the sun’s disk, while moving from B' to c'in its orbit, but during this time the 
observer is carried from b to e, by the diurnal motion of the earth, so that Venus must proceed to E' 
in its orbit before the observer at c can see it pass off from the sun’s disk at E, the length of the 
transit being increased by the motion of the observer in the ratio of B'e' to B'E'. From this differ¬ 
ence between the computed and observed durations the parallax is ascertained in the same manner 
as before. If now there be any error in the computed duration of the transit, the results of these two 
operations will be found unequal, ‘since any change in the computed duration increases one, while it 
diminishes the other, so that the mean between the two results (and in fact the mean of many has 


204 


OF ASTRONOMY. 


Book VII. Part I. 


Plate 10. 
Fig. 17. 


Plate II. 
Fig. 3. 


Plate 11 
Fig. 4. 


been taken) must in all probability give very correctly the sun's horizontal parallax on the day of 
the transit. Whence the sun’s horizontal parallax, at the time of his mean distance from the earth, 
may be found ; for (by Prop. XXIX.) as the sun’s mean distance from the earth is to his proportional 
distance at the transit, so is his horizontal parallax at that time to his mean horizontal parallax. 

In this manner the sun’s mean horizontal parallax has been found, from comparing the transits 
of Vonus in 1761 and 1769, to be 8.65" or about seconds. See Phil. Trans. Vol. LX1I. p. 611, and 
Ferguson’s Astronomy, Chap. XXIII. 

Schol. The transits of Venus happen but very seldom; the first that seems to have been noticed 
was in the year 1639, by Mr. Horrox and his friend Mr. Crabtree. With a view of engagingthe 
attention of future astronomers to the above method of determining the sun’s parallax, and thereby 
his real distance from the earth, Dr. Halley communicated a paper to the Royal Society in the year 
1691, containing an account of the several years in which such a transit would happen. He particularly 
mentioned those which would be seen in 1761 and 1769, presuming that on those periods this impor¬ 
tant problem would be solved with great accuracy. No other transit will happen until the year 1874. 

Except such transits as these, Venus exhibits the same appearance to us regularly every eight 
years; her conjunctions, elongations, and times of rising and setting, being nearly the same, on the 
same days as before. 

PROP. CXXXVII. To find the distance of the sun from the earth. 

In the triangle AHC, suppose H to be the sun. As the sine of 8|- seconds, the horizontal parallax 
of the sun AHC, is to radius, so is the semidiameter of the earth AC, which is found by mensura¬ 
tion to be 3956 English miles, to the number of semidiameters of the earth contained in the distance 
of the sun from the earth. Hence the sun’s distance from the earth is found to be about 95,173,000 
English miles; for by log. we have 5.621914 (sine of 8Y.65) : 10.000000:: 3.600428 (log. of 3956) 

: 95,173,000 miles. 

PROP. CXXX\ III. To measure the distance of Mercury or Venus from the 
sun. 

Let S be the sun, E the earth, and M Mercury or Venus. Measure the angle SEM, and observe 
accurately the time when this measure is taken. When mercury has made one revolution, and arrives 
at the same point M, the earth will be in some other part of its orbit, as R ; measure at that time the 
angle SRM, and observe the time when the measure is taken. 

By these two observations the time in which the earth passes from E to R is known; hence, as 
1 year is to the time employed in passing from E to R, so are 360 degrees to the arc ER; whence 
the arc ER, and the angle ESR, are found. In the triangle ESR the sides SE, SR, (the distance of 
the sun from the earth) being known, and also the contained angle RSE, let the angles at the base 
SER, SRE, and the base RE be found. Then from the known angle SER take away the angle SEM, 
which is also known, there will remain MER; and from the known angle SRE take away the known 
angle SRM, there will remain MRE. The two angles MER, MRE, being thus found, the third angle ' 
RME is also known ; and the side RE is known. Wherefore, the sine of the angle RME is to the side 
RE, as the sine of the angle MRE is to the side ME, or as the sine of the angle MER is to the side 
MR. In the triangle SRM, the sides RS, RM, being thus found, the sum of the two sides RS, RM, is 
to their difference, as the tangent of half the sum o‘f the angles at the base RSM, RMS, is to the tan¬ 
gent of half their difference. To half the sum add half the difference, and the greater anode at M.is 
found ; and from half the sum take away half the difference, and the less angle at S is found. Whence, 
the sine of the angle at M is to the side RS, or the sine of the angle at S is to the side RM, as the sine 
of the angle at R is to the base SM, which is the distance required. 

PROP. CXXXIX. To measure the distance of Mars fr om the sun. 

• 

Let S be the sun, E the earth, and M Mars. Measure the angle SEM; when Mars has made one 
revolution, observe the place of the earth in its orbit R, and measure the angle SRM. Having- found 
as before the arc ER, and the angle ESR, in the triangle ESR, in which the two sides SE, SR, and 
the contained angle ESR, are known, let the angles at the base SER, SRE, and the base RE, be found. 

If from the angle SEM (which has been observed) be taken SER, there remains REM ; and if from 
the. angle SRM (which has been observed) be taken SRE, there remains ERM. Whence, in the tri¬ 
angle RME, the angles at R and E being found, the third angle is known ; and the sine of the ano-le 
at M is to the side RE, as the sine of the angle at E is to the side RM. Wherefore, in the tri¬ 
angle SRM, the two sides of which, RS, RM, and the contained angle at R, are known; the two 
angles at the base, S, M, and lastly the base SM, which is the distance required, may be found. 


205 


Chap. IX. OF PARALLAXES, &c. 

PROP. CXL. To measure the distance of Jupiter or Saturn from the sun by 
their satellites. 

Let S be the sun, E the earth, and I Jupiter. First, observe the instant in which the satellite R riate 11 . 
disappears behind Jupite , and t ie instant in which it again appears; then, dividing the intermedia Fig.,5. 
ate time into two equal parts, this will g>ve the instant in which the earth E, Jupiter 1, and the 
satellite R, are in one right line LID. Next, observe the instant in which the satellite disappears 
behind the shadow of Jupiter, and the instant in which it again appears; and divide the time be¬ 
tween these instants into two equal parts to rind the instant in which the satellite is in the midst of 
the shadow, that is, in which the sun, Jupiter, and the satellite form a right line SIR. lienee, the 
time taken up in passing from D to R is known ; whence, the time of the entire revolution of the 
satellite is to 360 degrees as the time employed in passing from D to R is to the arc DR. Thus the 
arc DR, and the angles RID, EIS are found. Lastly, having taken an observation of the angle IES, 
the other angle ESI is found; and the side ES, the earth’s distance from the sun, is known; whence, 
the sine of the angle Eld is to the side ES, as the sine of the angle IES is to the side IS, the distance 
required. 

PROP. CXLI. To measure the distance of any planet from the sun. 

Because the real distances of the planets from the sun are as their proportional distances; as the 
proportional distance of the earth from the sun is to the proportional distance of any other planet from 
the sun, so is the real distance of the earth from the sun in miles, to the real distance of any other 
planet from the sun in miles. 

Hence are found the distances of the planets from the sun in English miles. Mercury, 36,841,468; 

Venus, 68,891,486 ; Mars, 145,014,148; Jupiter, 494,990,976 ; Saturn, 907,956,130; and the Herschel, 
1,800,000,000. 

PROP. CXLII. The horizontal parallax of any planet being given, to find its dis¬ 
tance from the earth. 

Let H be the planet, whose horizontal parallax AHC is known. The semidiameter of the earth_AC 1>la te 10 . 
being known, in the triangle CAH the sine of the angle AHC is to the side AC, as the sine of the angle * lg ‘ 
HAC is to the side HC, the distance sought. 

PROP. CXLIII. The distance of any planet being given, to measure its real mag¬ 
nitude. 

Let A be the earth and C the centre of any planet; and let the distance CA be known. Suppose Plate 11 . 

two right lines, AB, AD, drawn tangents to the planet, the angles CBA, CDA, are right angles ; there- Fig. 6 . 

fore the square of AC is equal to the two squares of AB and BC together. The same square of AC 
will also be equal to the two squares of AD and CD And since the square of the radius CB is equal to 
the square of the radius CD (on account of the spherical rigure of the planets), the square of the tangent 
AB is equal to the square of the tangent AD, and the tangent AB to the tangent AD. Hence the two 

triangles ABC, ADC, are equal, and consequently the angles BAC, CAD, are equal. The angle BAD be¬ 

ing measured by a micrometer, its half BAC is known ; whence, in the triaqgle ABC, the sine of the 
angle at B, which is a right angle, is to the side AC, as the sine of the angle at A is to the side 
BC. The radius, and consequently the diameter of the planet, being thus found, because spheres are 
as the cubes of their diameters, its magnitude is known by rinding the cube of its diameter. 

Scitol. In Plate 15. Fig. 4, we have a view of the proportional magnitudes of the planets Mercury, p [ate ( . 
Venus, the Earth and Moon, Mars, Jupiter, and Saturn, according to Ferguson; with the addition of p;g, 4 
HerscheL In proportion to these figures of the planets, the sun’s diameter is about two feet. 

PROP. OXLIV. To find the periodical time of a planet. 

Because, whilst any planet is performing its revolution, the earth is carried forward in its path, the 
planet, after one greatest elongation, must not only complete a revolution, but likewise the whole 
angular space which the earth described in that time, before it arrives again at the same elongation. 

Thus, before Venus can return to the same elongation, besides performing an entire revolution in its 
orbit (equal to 4 right angles), it must pass through as much more angular space, as the earth has done 
in the mean time. Hence, as the angular motion of Venus is to the angular motion of the earth in the 
time between the greatest elongation and its return, so is the periodical time of the earth to the peri- 
tical je of Venus. In this manner the periodical times of all the planets may be found. 


OF ASTRONOMY. 


Book VII. Part I. 


Or, observe when a planet is in any point of its orbit, and, after any number of revolutions, observe 
when it comes to the same point again ; then divide that interval of time by the number of revolutions, 
and you get the time of one revolution. The observations of ancient astronomers are here very use¬ 
ful; for as they have put down the places of the planets from their own observation, by comparing 
them with the places observed now, we take in a very great number of revolutions, and, therefore, if we 
divide the interval of time by the number of revolutions, should a small error be made in the whole 
time, it will affect so much less the time of one revolution. The periodical times of the planets will be 
found in the table at the end of the chapter. 

Con. Because the squares of the periodical times of the planets were found by Kepler to be as the 
cubes of their distances, the periodical times of any two planets being known, and the comparative or 
real distance of one of them from the sun being given, the distance of the other may from this propor¬ 
tion be found. 

PROP. CXLV. To find the mean velocities of the planets. 

The periodical time of a planet being known, and also its diameter, and consequently its circumfer¬ 
ence (for the diameter of a circle is to its circumference nearly as 113 to 355), its mean velocity, or 
the velocity with which it wou.d move if its motion were uniform, may be thus found ; as the whole 
periodical time of the planet is to an hour, so is the whole circumference of its orbit to the angular 
space passed over in an hour. Thus it is found, that the mean horary velocity of the earth is 68216.9 
English miles. In like manner, the horary velocity of the other planets may easily be found. 

Cor. By comparing this proposition with Cor. Prop. CXVII. the velocity of light wiil be found to 
be to the velocity of the earth in its orbit as 10632 to 1. 

PROP. CXLVI. The planets revolve round their axes. 

It is found by observation, that the earth revolves about its axis in 23h. 56' 4" mean solar time; Sat¬ 
urn in 12h. 13'£; Jupiter in 9h. 56'; Mars in 24h. 40'; Venus in 23h. 20'. The sun is found to rfevolve 
on his axis in 25d. lOh. 

The time of Saturn’s rotation is computed from the ratio of its diameters, which Dr. Herschel makes 
to be about 11 to 10. The time of the rotation of the other planets is not known ; nor has it yet been 
determined whether they do revolve about their axes. 

Schol. The following Table contains a synopsis of the distances, magnitudes, periods, &ic. of the 
several planets, according to the latest observations. 


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BOOK VII. PART II 


Plate 11 
Fig. 7. 


OF THE CAUSES OF THE CELESTIAL MOTIONS AND OF OTHER 

PHENOMENA. 


CHAPTER I. 

Of the Cause of the Revolutions of the Heavenly Bodies in their Orbits. 

PROP. CXLVII. The moon is retained in its orbit by a force which impels it to¬ 
ward the centre of the earth. 

Since (by Book II. Prop. I.) the moon, or any other planet, being put into motion, would continually 
move on uniformly in a right line, there must be some force which draws it from its rectilineal path. 
Whatever this force is, since it is found by observation that the moon by a radius drawn to the earth’s 
centre describes equal areas in equal times, it follows (from Book II. Prop. LXX11I.) that it is impelled 
by that force toward the earth’s centre. The earth indeed is not at rest; but because both the moon 
and earth revoive round the sun, the motion of the moon with respect to the earth is the same as if the 
earth were at rest. 

PROP. CXLVIII. The force which retains the moon in its orbit is, at different 
distances from the earth, inversely as the squares of those distances. 

The mcon’s orbit being an ellipse, which has the earth in one of its foci, the force which retains it 
in its orbit must (by Book II. Prop. LXXXI.) in different parts of the orbit be inversely as the squares 
of the distances from the earth. 

PROP. CXLIN. The moon is retained in its orbit by a force which carries it to¬ 
ward the earth with the same velocity with which a body, acted upon by gravitation 
at the distance of the moon, would fall toward the earth. 

Let AbB be the earth, PLV a part of the moon’s orbit, LC an arc which the moon describes in its 
orbit in one minute of time. Since the moon describes its whole orbit in 27 days 7 hours 43 minutes 
that is, in 39343 minutes, the length of the arc LC, which the moon describes in one minute, is the 
P art 360°, or 33 "- If the moon, setting out from L, were not impelled toward the earth, it 
would move in the right line LB. Since therefore it moves in the arc LC, there must be a force im¬ 
pelling it toward the earth’s centre which draws it from the tangent LB, so that, at the end of I minute 
when it is arrived at C, it will have departed from the tangent as far as BC, or LD ; or, because the 
moon describes the diagonal LC in 1 minute, it would in the same time, by the projectile force, de 
scribe LB, and by the centripetal, LD. LD is then the space through which the centripetal force makes 
the moon fall toward the earth in 1 minute; and LD is the versed sine of the arc LC, which is an arc 
of 33 '. Therefore the force which impels the moon, would make it fall, in 1 minute of time, through 
the versed sine of an arc of 33". Because AER, the earth’s circumference, is found to measure 
123249600 Paris feet, its semidiameter AT will be about 19615300 such feet. Since therefore the 
mean distance of the moon from the earth is found to be 60 semidiameters of the earth, AT multiplied 
by 60 will give the length of LT, a semidiameter of the moon’s orbit, namely, 1176943000 feet. And 
as the radius is to the versed sine of 33", so is LT to LD, or nearly as 1176948000 to 15 yL Paris feet, 
which is nearly equal to 16§ English feet, or 1 pole. Therefore, if the moon were to fail toward the 
earth, the centripetal force which impels it toward the earth would make it fall 1 pole in the first min¬ 
ute of its descent. But because (by Prop, CXLVIII.) this centripetal force is inversely as the squares of 
the distances, a body which is at the distance of the moon, or 60 semidiameters of the earth, will be 




% 


Chap. I. CAUSE OF CELESTIAL MOTIONS. 

attracted by a force as much less than that at the surface, as the square of GO, or'3G00, is greater than 
the square of 1, or 1 ; that is, the force at the surface being 1, it will be to the force at the distance of 
the moon, as 1 to^ an( l the velocities will have the same ratio. But a body at the surface of the 
earth falls through 1 pole in a second of time; that is, (by Book II. Prop. XXVI.) through 3600 poles 
in a minute. Therefore, at the distance of the moon the body would fall through part of this 
length, that is, through 1 pole in a minute. But it has been shown that the moon, by its centripetal 
force, falls toward the earth 1 pole in a minute ; therefore the moon is retained in its orbit by a force 
which moves it with the same velocity with which a body, acted upon by gravitation, and removed to 
the distance of the moon from the earth, would fall toward the earth. 

PROP. CL. The moon is retained in its orbit by the force of gravitation. 

The force which retains the moon in its orbit agrees with gravitation in its direction (by Prop. 
CXLVII.) and in its degree of force (by Prop. CXLIX.) ; it may therefore be concluded to be the 
force of gravitation. 

PROP. CLI. The primary planets are retained in their orbits by a force which 
impels them toward the sun. 

It is found by observation, that each of them, as they revolve in their respective orbits, describes^ 
by a radius drawn to the sun, equal areas in equal times. Therefore (by Book II. Prop. LXXI1I.) the 
force which retains them in their orbit, impels them toward the sun. 

PROP. CLII. The forces which retain the primary planets in their respective 
orbits are, at different distances from the sun, inversely as the squares of those 
distances. 

It appears from observation, that the squares of the periodical times of the planets are as the cubes 
of their mean distances from the sun. For example, Saturn’s periodical time being found to be to 
Jupiter’s about as 30 to 12, and the distance of the former from the sun, to that of the latter nearly 
as 9 to 5 ; the squares of the times are 900 and 144; which are to one another nearly as 729 to 125, 
the cubes of the distances. This proportion takes place in all the primary planets ; hence (by Book II. 
Prop. LXXIX.) the force by which they are retained in their orbits is inversely as the squares of the 
distances. 

Schol. It has been remarked in the last chapter, that Kepler found by observation that the squares 
of the periodical times’of all the primary planets are as the cubes of their mean distances from the 
sun. Astronomers have since found, that the same law holds good in the secondaries with respect to 
their primaries. To Sir I. Newton we are indebted for an investigation of this law on physical prin¬ 
ciples. He has demonstrated, that, in the present state of nature, such a law was inevitable. 

PROP. CLIII. The primary planets are retained in their orbits by the force of 
gravitation. 

The moon having been shown from the direction, and the law of its centripetal force, to be re¬ 
tained in its orbit by gravitation ; since the primary planets are impelled toward the sun (by Prop. 
CLIA as the moon is toward the earth, and since their centripetal force acts with respect to the sun 
bv the same law, by which the force which retains the moon in its orbit acts with respect to the 
earth, namely, that this force is inversely as the square of the distance of the planet from the sun; 
it may be concluded, as in the case of the moon, that they are retained in their orbits by the force 
of gravitation. 

This follows likewise from their moving in elliptical orbits, since it has been proved (Book II. 
Prop. LXXXI.) that bodies revolving in such orbits have their centripetal forces inversely as the 
squares of their distances from the centre about which they revolve. 

PROP. CLIV. The satellites of Jupiter and Saturn are retained in their respect¬ 
ive orbits by the force of gravity. 

They are observed to describe equal areas round the respective primaries in equal times, and con¬ 
sequently (by Book II. Prop. LXX111.) are impelled toward them; and the forces which retain them in 
their orbits are at different distances inversely as the squares of those distances (by Book II. Prop. 
LXXXI.), because it has been observed that the squares of their periodical times are as the cubes of their 

27 


209 


210 


OF ASTRONOMY. 


Book VII. Part II. 


distances from their respective centres. Therefore the force which retains the satellites of Jupiter and 
Saturn in their orbits acts in the same manner, and by the same law, as the force which retains the moon 
in its orbit acts with respect to the earth. But all effects of the same sort are, without proof of the 
contrary, to be considered as produced by the same cause. Therefore the power which retains the 
satellites in their orbits is gravitation. 

PROP. CLV, The sun and any planet revolve round a common centre of gravity, 
which remains at rest. 

riate 11, Let S be the sun, and P any planet, mutually attracting each other. If neither of the two bodies 

Fig. a. revolved in any orbit, they w'ould move toward each other, and would meet at C, their common 
centre of gravity ; and during the approach of these two bodies, C, their common centre of gravity, 
W'ould be at rest, (by Book II. Prop. LI.) But if the body P have a projectile force given to it in the 
direction P t, and if this projectile force and its gravitation toward S make it describe an orbit round 
S, (by Book II. Prop. LXVIII.) such a projectile force will prevent the body P from approaching to 
S, though it gravitates toward S. But if S has not as great a projectile force given to it at the same 
time in the opposite direction S s, then because S continues to gravitate tow ard P, and there is no 
force which can prevent its approaching to P, it follows that S will approach to P, or, as P revolves 
round S, the mutual gravitation of these two bodies will diminish the distance SP. Now it appears 
(from Book II. Prop. LI.) that C, the common centre of gravity, always divides this distance SP in the 
inverse ratio of S to P, or that SC is always as much less than PC, as the quantity of matter in S is 
greater than the quantity of matter in P; consequently, since the quantity of matter in S and in P is 
always the same, SC and PC have always the same ratio to one another. But as S approaches to P, 
SC decreases. Therefore PC must decrease in the same ratio. But PC can decrease no otherwise 
than either by the approach of P to C, or by the approach of C to P. But the projectile force pre¬ 
vents P from approaching to C. Therefore C must approach to P. Thus it appears that, if P has a 
projectile force given to it, and is made to revolve, unless S has an equal projectile force given to 
it at the same time, the mutual gravitation of these two bodies toward each other will put C, which 
is their common centre of gravity, in motion ; contrary to Book II. Prop. LI. Cor. Therefore as the 
planet P begins to move in the direction P t, the sun S will likewise begin to move in the direction 
Ss; and C, their common centre of gravity, will continue at rest. And as they tend mutually toward 
each other, or toward C, their common centre of gravity, their motions will not continue to be per¬ 
formed in right lines, but the planet P will revolve round C in an orbit, of which PR is a part, and 
the sun S will revolve round C in an orbit, of which SQ, is a part. 

PROP. CLVI. The sun and any planet, whilst they mutually gravitate toward 
each other, describe similar figures round their common centre of gravity, and 
round each other. 

Plate 11. Let S be the sun, P the earth, or any other planet, and C their common centre of gravity, about 

* c - which (by the last Prop.) they revolve. To a spectator at P, who imagines the planet to be at rest, the 
sun will appear to revolve about P, and the reverse at S. Because the common centre of gravity of the 
sun S, and any planet P, is alw ays in a right line drawn from the sun to the planet, if the planet moves 
through any small space from P top, the line p C continued must pass through the sun; and conse¬ 
quently the sun must have moved from S to s. Thus Pp, Ss, are arcs described by the planet and 
sun in their respective orbits in the same time, and PC p, SC s, are areas described in the same time by 
the radii CS, CP. \nd because the vertical angles at C are equal, and SC is to PC. as s C to p C 
(for SP, s p, are both divided in C in the inverse ratio of the quantities ot matter in P and S) the areas 
PC p, SC s, are similar. In like manner, any other parts of the two orbits described in the same time 
may be shown to be similar; consequently, the whole orbits are similar. 

Again, when P has completed its revelation round C, or or i of its orbit, it will appear to a 
spectator at S, to whom S seems at rest, to have completed its orbit, or i or i of its orbit round S. 
And universally, the angular motion of the planet P about C, in any given time, will be equal to its 
apparent angular motion about S, considered as at rest by a spectator at S. If therefore the planet 
P in any given time has moved from P to p, in which (by the last Prop.) the sun S has moved from 
S tos, the angle PCp, which is the measure of the planet’s angular motion about C, will be equal to 
the apparent angular motion rounds. Let St be taken equal to sp, and make the angle PS t equal 
to the angle PC p; P will, by a radius drawn to S, apparently describe the area PS t, whilst by a radius 
drawn to C, it is describing the area PC p. Now, because (as was before shown) SC is to PC, as s C to 
p C, (El. V. 18.) SC -f CP, or SP, is to PC, as s C -f- Cp, or sp is to pC; and sp is equal to St; 
therefore PS is to PC as S t is to pC. Consequently, the two figures PC p, PS t, are similar. In 
like manner it may be shown, that any other part, described in any given time, of the orbit the planet 


Chap. LI 


OF THE LUNAR IRREGULARITIES. 


213 


appears to move in round the sun considered as at rest, will be similar to other parts, described in the 
same time, of the orbit in which the planet moves round the common centre of gravity of the sun and 
the planet; therefore the whole orbits are similar. And since the orbits which the sun and the plan¬ 
et describe about their common centre have been proved to be similar, it follows, that the orbit which 
any planet appears to describe round the sun, considered as at rest, is similar to the orbit which the 
sun in the mean time describes round the common centre of gravity. 

In like manner it might be proved, that the orbit which the sun S appears to describe round the 
planet P, considered as at rest, is similar to either of the orbits which the planet and sun describe 
about the centre of gravity. 

Cor. If the sun’s apparent motion, seen from the earth, be an ellipse, having the earth in one of 
its foci, the earth’s apparent motion, seen from the sun, will be in a similar ellipse, having the sun in 
one of its foci; and if the sun and earth mutually gravitate toward each other, they describe similar 
elliptic orbits about their common centre. 

PROP. CLVII. The common centre of gravity of the sun and all the planets is 
at rest, and is the centre of the solar system. 

Since, from the mutual gravitation of the sun and any one planet, they will revolve about their 
common centre (by Prop. CLV.), the same must hold good with respect to the sun and all the planets. 
Consequently, there must be some one point in the solar system which is its centre of gravity, and 
is at rest. 


CHAPTER II. 

Of the Lunar Irregularities. 

PROP. CLVIII. The nearer the moon is to its syzygies, the greater is its veloci¬ 
ty ; and the nearer it is to its quadratures, the slower it moves. 

Let S represent the sun, T the earth, and LMNO the orbit of the moon ; let the moon be in one Plale 11. 
of its quadratures at L, and let the lines LS and TS be drawn. It is obvious, that the tendency which Pig- 8 * 
the moon has toward the sun is along the line LS, and that which the earth has, is along the line TS; 
let then the former of these be resolved into two others, the one along LA parallel and equal to TS, 
the other from L to T, along the line LT. The former of these tendencies being parallel and equal to 
that bv which the earth tends along the line TS, alters not the situation of the two bodies L and T 
with respect to each other, that is, it disturbs not the motion of the body L; but the other along LT 
increases its tendency toward T. And this increase will be to the tendency the moon has to A, which 
is the same which the earth has.to S, as the distance LT to LA, or TS; that is, the gravity of the moon 
toward the earth in the quadratures is augmented by the action of the sun, and that augmentation is to the 
tendency which the earth has to the sun, as the length of the line LT, or the distance of the moon 
from the earth, to TS, the distance of the earth from the sun. 

Hence the greater the moon’s distance is from the earth, the distance of the sun remaining the same, 
the greater will this increase of the moon’s gravity toward the earth be. But if the distance of the 
moon from the earth remain the same, and the distance of the sun be augmented, this additional in¬ 
crease will be the less in the ratio of the cube of that distance. For, if TS be increased while LT 
remains the same, LT will be so much the less with respect to TS, that is, the increase will be diminish¬ 
ed in the ratio of the sun’s distance ; but when TS, the distance of the sun, is increased, the absolute 
force of the sun, and with it the abovementioned increase, will be diminished also in proportion to the 
square of that distance; consequently, taking in both considerations, it will upon the whole be diminish¬ 
ed in the ratio of the cube of that distance. 

Let now the moon be in one of its syzygies at M, then will the tendency it has to the sun, more than 
that which the earth has, which is farther off at T, be to that which the earth has, as the difference of 
the squares of SM and ST is to the square of Sl\I; but the difference between the squares of SM and 
ST has nearly the ratio to the square of SM, which twice MT, that is, MO, has to SM ; because the 
difference between the squares of two numbers whose difference is very small with respect to either 
of them (as the difference between SM and ST is with respect to the distance of S) has little 
more than double the ratio to the square of the less number, that the difference between the num¬ 
bers themselves has to the less number. The tendency therefore which the moot., wnen at M, has 
to the sun,- more than that which the earth has, is to that which the earth has, nearly as MO, or twice 



212 


OF ASTRONOMY. 


Book VII. Part II. 


TL, to SM, or, because of the sun’s great distance, as twice LT to TS. Her tendency therefore to the 
earth is now diminished in that ratio ; but, as was shown above, it was augmented in the quadratures in 
the ratio only of LT to TS. The diminution here is therefore nearly double of the augmentation there. 
And whereas that augmentation, when the distance of the sun remains the same, was sbowm to increase 
with the distance of the moon; but when the distance of the moon remains the same, to decrease with 
the cube of the sun’s distance; this diminution, being always nearly double of that, will do the same. 

When the moon is in the other syzygy at O, it is attracted toward the sun less than the earth is by 
the difference of the squares of SO and ST; which, in effect, is the same thing as if the earth were 
not attracted at all toward S, and the moon were attracted the contrary way; so that its tendency to 
the earth is here also diminished, as weli as when it was at M, and almost in the same degree ; fox', on 
account of the sun’s great distance, the diffeience betw’een the squares of SO and ST is nearly the same 
as between ST and SM. 

Or thus ; the annual course of the moon round the sun being performed in the same time that the 
earth’s is, it ought to be retained in that course by the same force that the earth is; whereas, when the 
moon comes to M, the action of the sun upon it is greater than it is upon the earth, by the difference of 
the squares of SM and ST; and when the moon is at O, it is less than it is upon the earth by the differ¬ 
ence between the squares of ST and SO ; so that in the former case the moon is drawn too much toward 
the sun, and in the latter too little ; and therefore in both cases its tendency toward the earth is diminish¬ 
ed, and almost in the same degree ; because, as was observed above, the difference of the abovementioned 
squares is nearly the same in either case. 

Next, let the moon be in the point of her orbit between the quadi'ature and the syzygy. Then being 
nearer the sun than the earth is, she will be attracted with a stronger force ; let it be expressed by 1 5 
produced to D, till l D be of such a length, that TS being put to expi’ess the action of the sun upon the 
earth, ID may express the stronger force of the sun upon the moon; and let ID be resolved into two 
others, one of which let be Za, equal and parallel to TS, then will the other be a D, or its equal and 
parallel ID. This IG is the only disturbing force upon the moon at L, the other La being parallel to 
TS, affects the moon just as the sun does the earth ; and thus alters not their situations with respect to 
* each other. Let then this figure with the line LG be transferred to fig. 9. This force LG may be re¬ 
solved into LI and LH, the one a tangent to the orbit of the moon, and the other a perpendicular there¬ 
to ; the former accelerates the motion of the moon when going from the quadrature at O to the syzygy 
at B; and will retard it when going from B to R. The other, when H falls upon TL produ ed, as in 
this figure, diminishes the tendency of the moon toward the earth ; and when it falls between L and T. 
it augments it. 

Thus the nearer the moon is to its syzygies, the greater will be its velocity; and the nearer it is 
to the quadratures, the slower it will move; because one of the forces into which LG is resolvable, 
accelerates its motion from the quadrature to the syzygies; and retards it as much from thence to the 
quadratures. 

Cor. Hence the moon in her monthly revolution is, by the action of the sun, alternately accelerated 
and retarded. 

PROP. CLIX. The moon describes equal areas in equal times only at the syzygies 
and quadratures, and deviates from this law the farthest in the octants. 

Plate 11. The disturbing force being resolved into two others, one of them, at the quadratures, or syzj'gies, 
Fig. 9. w jll be found to point from or toward T the centi’e of the earth directly, and therefore will not hinder 
the moon from describing equal areas in equal times; the other likewise, in those places, will be found 
to tend toward the centre of the sun, and therefore neither of them will prevent the moon there from 
describing equal areas in equal times, that is, will not at the quadratures disturb the moon’s motion 
at all. 

But when the moon is in the octants, as at L, this force being resolved into two others, one of them, 
as L1I, will point directly to or from the centre of the earth, and therefore will increase or diminish 
the moon’s tendency toward the earth, but not hinder it from describing equal areas in equal times. 
But the other, as LI, or HG, points neither toward the centre of the earth, nor sun, and therefore, in 
the octants, prevents its describing equal areas in equal times. But this being the mid-way between 
the quadrature and the syzygy, in both which places this disturbing force doth not prevent the moon 
from describing equal areas in equal times, it follows, that at the octants, this disturbing force will be 
greatest of all. 

Schol. Hence it has always been found more difficult to obtain the moon’s place in the octants, so as 
to agree with observation, than at the quadratures or syzygies 


i* 


Chap. II. 


OF THE LUNAR IRREGULARITIES. 


213 


PROP. CLX. The orbit of the moon is more curved in the quadratures, and less 
in the syzygies, than it would be if it were only attracted by the earth. 

For its motion (by Prop. CLV ill.) being accelerated during its progress from the quadratures to 
the syzygies, in the syzygies its motion will be quicker than it ought otherwise to be, and there¬ 
fore its centripetal force less than it would otherwise be. It will therefore at the syzygies describe 
the portion of a larger curve, which consequently will be less curved than a smaller. On the other 
hand, while the moon passes from the syzygies to the quadratures, its motion is continually retarded, 
and therefore, at the quadratures, its motion will be slower than it would otherwise be. At the quad¬ 
ratures, therefore, the moon will describe the portion of a lesser curve, which therefore will be more 
curved than a larger curve. 

PROP. CLXI. When the earth is in its perihelion, the periodical time of the moon 
will be the greatest; when the earth is in its aphelion, the periodical time of the moon 
will be the least. 

Since the irregularities explained in the three preceding Propositions proceed from the action of 
the sun, it follows, that where the action of the sun is greatest, the irregularities arising from it will be 
greatest too. But the nearer the earth is to the sun, the greater will be the action of the sun upon the 
moon; and the more the moon tends toward the sun, the less will it tend toward the earth. When pi a t e n. 
therefore the earth is at the perihelion A, and consequently at its least distance from the sun S, the ac- Fig. 10. 
tion of the sun upon the moon will be greatest, and destroy more of its tendency toward the earth than 
at any other distance, as SE, SC, SD, &c. Therefore when the earth is at the perihelion A, the moon 
will describe a larger orbit about the earth, than when the earth is at any other distance from the sun, 
and consequently her periodical time will then be the longest. 

But the earth is at its perihelion in the winter, and, consequently, then the moon will describe the 
outermost circle about the earth, and her periodical time will be the longest; which agrees with obser¬ 
vation. For the same reason, when the earth is at its aphelion B, the tendency of the moon toward the 
earth will be greatest, and consequently her periodical time the least. And in this case, which will be 
in the summer, it will describe the innermost circle about the earth. 

PROP. CLXII. The line of the moon’s apsis goes forward when the moon is in 
S yzygy, and backward when it is in quadrature ; but it goes farther forward than 
backward each time, so that at length it performs a revolution according to the order 
of the signs. 

Since the moon describes an elliptical orbit CEDF about the earth S, placed in one of its foci , and 
since its centripetal force toward the earth, by means of the action of the sun (by Prop. CLVIII.) is pi* te jQ 
continually increasing, or decreasing, but not equably ; that is, sometimes less, and sometimes more, °‘ 
than in the inverse duplicate ratio of the distance of the moon from the earth, therefore the line of 
the moon’s apsis AB will be continually going backward or forward ; that is, the axis AB will not al¬ 
ways lie in that situation, but go backward into the situation CD, or forward into the situation EF. 

Since however, taking one whole revolution of the moon about the earth, the action of the sun more 
diminishes the tendency of the moon toward the earth than it augments it, therefore the motion of the 
apses forward exceeds their motion backward. Upon the whole, therefore, the apses of the moon’s or¬ 
bit go forward, or according to the order of the signs. Their revolution is completed in about 9 years. 

PROP. CLXIII. The eccentricity of the moon’s orbit is varied in every revolution 
of the moon, and is greatest when the moon is in syzygy, and least when it is in quad¬ 
rature ; and the orbit is most of all eccentrical when the line of the apsis is in the sy¬ 
zygies, and least of all eccentrical when this line is in the quadratures. 

Because the moon describes an eccentrical orbit CEDB about the earth S, a id the action of the Plate 11. 
sun upon it sometimes increases its tendency toward the earth, and sometimes diminishes it, that is, Fig- 10 . 
makes its gravity toward the earth increase or decrease too first; if, while the moon ascends from its 
lower apsis A, its gravity toward the earth decrease too fast, instead of describing the path DBF, and 
coming to the higher apsis at B, it will run out into a curve beyond DBF; that is, the orbit will become 
more eccentric, or farther from a circle. On the other hand, if the moon is passing from her higher 
apsis B, to her lower A, and its gravity toward the earth, by the action of the sun, increase too fast, 
it will approach nearer to the earth than the curve CAE, and describe a curve within CAE, or a por¬ 
tion of an orbit less eccentric, or nearer to a circle, than CEDF. And if we compare several revolu- 


214 


OF ASTRONOMY. 


Book VII. Part IL 


Plate 11. 
Fig. 11. 


Plate 11 
Fig. 12. 


lions of the moon together, we shall find that when the line of the apsis is in the syzygies, the eccem 
tricity will be the greatest of all, because in that situation, the difference between the tendency which 
the moon has to the earth in one of the apses, and that which it has in the opposite one, is the great¬ 
est of all; whereas, when the. line of the apsis is in the quadrature, this difference is the least, and 
therefore the lunar eccentricity will be so too. 

Cor. When the gravity of the moon toward the earth decreases too fast, the eccentricity of her 
orbit will increase ; and when her gravity toward the earth increases too fast, the eccentricity of her 
orbit will decrease; and the orbit itself will approach nearer to a circle. 

PROP. CLXIV. The line of the nodes moves backward, but not uniformly; when 
it is in the syzygies it stands still, ami moves fastest in the quadratures. 

When the line of the nodes is in the syzygies, as CD, the plane of the moon’s orbit passes through 
the centre of the sun S, as well as through that of the earth E; whence, the disturbing force acting 
in the direction of the line of the nodes, and consequently in the plane of the lunar orbit, the moon 
is not drawn out of the plane of its orbit by the sun. But when the line of the nodes is in any other 
situation, and the moon not in one of the nodes, it is continually drawn out of the plane of its own or¬ 
bit, on that side on which the sun lies. For instance, if the plane of its orbit CGDF produced passes 
above the sun, the sun draws it downward; if, on the contrary, the plane of its orbit produced passes 
below the sun, it draws it upward. Hence it follows, that when the line of the nodes is notin the syzy¬ 
gies, and the moon, having passed either of the nodes, has got out of the plane of the ecliptic ACBD, 
on either side of it, the action of the sun occasions the moon to return back to the plane of the ecliptic 
sooner than it otherwise would do. But where the moon enters that plane, there is the next node ; 
so that each node does, as it were, come toward the moon; and the nearer the line of the nodes is to 
the quadratures, the greater is this effect, because, in that case, the sun is the farthest of all from the 
plane of the lunar orbit produced. So that the line of the nodes goes backward the fastest of all, when 
it is in the quadratures; and not at all in the syzygies. 

PROP. CLXV. The inclination of the lunar orbit is liable to change, and is great¬ 
est when the nodes are in the syzygies, and least when they are in the quadratures. 

When the nodes are in the quadratures A, B, and the moon in its orbit AGBF has passed A, and 
is approaching the syzygy which is next to the sun, the action of the sun upon the moon prevents 
its ascending so high, that is, departing so far from the plane of the ecliptic ADBC; whence the in¬ 
clination of its orbit to the ecliptic will become less, and it will come to conjunction with the sun at 
H, making an angle with the ecliptic HAD, less than GAD. As tke moon goes on to the next quadrature 
B, the action of the sun upon the moon, in its descent toward the node, hastens its descent, and thus, 
blunging it down to the ecliptic at K sooner than it would otherwise arrive there, increases the in¬ 
clination of the plane of its orbit as much as it was diminished in ascending from A to H. And for 
the same reason, while the moon passes from B to the opposite syzygy F, the action of the sun decreas¬ 
es the inclination of its orbit, and increases it again on its passage from thence to A, the next quad¬ 
rature. 

When the nodes are in the syzygies, C, D, the plane of the moon’s orbit produced, passes through 
the centre of the sun ; and consequently, not beiug affected by the action of the sun, its inclination 
is neither increased nor diminished. 

But while the nodes are passing from the syzygies C, D, to the quadratures A, B, the inclination 
of the moon’s orbit is diminished in every revolution of the moon ; and while they are passing from 
thence to the syzygies, it is continually increasing. Suppose the nodes in the octants at O and L, and 
the plane of AGBF, the orbit of the moon, so inclined to ADBC, the ecliptic, that if produced it will 
pass above the sun S. When the moon is nearer the sun than the earth is, it is attracted toward the 
sun more than the earth is; and when farther off, the earth is attracted more than the moon is, that 
is, the moon is, as it were, attracted the other way. Hence, whilst the moon is ascending from the 
ecliptic in passing from O to P, the disturbing force being toward S, and the orbit above S, the moon 
will not rise so high as P, and the inclination of its orbit will be diminished while it is passing over 90 
degrees from the node O. In going from a point below P to the next quadrature B, which is 45 degrees, 
the disturbing force being still toward S, because the moon is as yet nearer the sun than the earth is, and 
the moon now descending toward the ecliptic, the attraction of the sun will hasten its descent, and there¬ 
fore cause it to move in a plane which will make with the plane of the ecliptic a larger angle than 
before; that is, in passing from P to B the inclination of the orbit is increased. But when the moon 
has passed B, and is moving toward L, the disturbing force acting, in the plane of the ecliptic, from 


Chap. II. 


OF THE LUNAR IRREGULARITIES. 


21 


the sun, and the moon still descending toward the ecliptic, the disturbing force, attracting the moon 
upward, will retard its descent to the ecliptic, and cause it to move in a plane which will make a 
less angle with the plane of the ecliptic than before; that is, while it is passing from B to the node 
L, the inclination of its orbit is diminished. Thus, while the moon passes from O to L, the inclina¬ 
tion of its orbit is diminished during three fourths of the passage. In like manner, while the moon 
is ascending from L to 1, because the disturbing force acts from the sun, the inclination of its orbit is 
diminished; and while it is descending from 1 to A, the disturbing force still acting from the sun, the 
inclination is increased. But while it is still descending from A to O, because the disturbing force acts 
toward the sun, tne inclination is diminished. Add to this, that while the moon passes from O to P, and 
from L to I, the disturbing force is much sweater than when it was passing from P to L, and from I to O, 
because the difference between the distances of the moon and of the earth from the sun is greater 
in the former case than in the latter. On the whole, therefore, while the nodes are between A and 
D, B and F; that is, while they are passing from syzygy to quadrature, the inclination of the lunar 
orbit is diminished; for, though the nodes have been supposed equally distant from the quadrature 
and syzygy, it is obvious that the like effects must happen, though different in degree, when they are 
nearer to the one than the other. 

Next, let the nodes be in the octants I, P, between A and F, and B and G. While the moon is as¬ 
cending from the node I toward the quadrature A, the disturbing force acting/rom the sun, it will be 
drawn upward, and the inclination of its orbit will be hereby increased. In ascending from A to O, 
the disturbing force acting toward the sun, its ascent will be diminished, or the inclination of its orbit 
lessened; but in descending from O to the node P, the disturbing force still acting toward the sun, it 
will be drawn downward, and, consequently, the inclination of its orbit will be increased. Thus, dur¬ 
ing one whole revolution of the moon in this position of its nodes, the inclination of its orbit will be 
increased through three fourths of its passage. And this will be true, as in the other case, when the 
nodes are not in the octants. Also, for the reason mentioned in the other case, the force which increas¬ 
es the inclination of the orbit is, while it acts, superior to that which diminishes it. While the nodes, 
therefore, are passing from the quadratures to the syzygies, the inclination of the moon’s orbit is in¬ 
creasing.' From all which it is manifest, that the inclination of the lunar orbit is the least when the 
line of the nodes is in quadrature, and the moon in syzygy, and greatest when the line of the nodes 
is in syzygy. 

PROP. CLXVI. The nodes of the moon are at rest, when the line of the nodes 
is in syzygy ; they move in antecede?itia , or from east to west, when the line of the 
nodes is in quadrature ; and also when it is between quadrature and syzygy; but 
their regress, in one revolution, is, in this case, less than when the line of the nodes 
is in quadrature. 

When the line of the nodes is in syzygy, because the disturbing force acts in the plane of the Plate II. 
moon’s orbit, it cannot change the inclination of that plane to the ecliptic; whence the common in- F, °- 12 - 
tersection of the two planes, or the line of the nodes, is immovable. If, whilst the line of the nodes is 
in AB, the moon is passing from A through G to B, being constantly drawn toward the plane of the eclip¬ 
tic by the disturbing force, it will come to the plane sooner than it would have done if no such force 
had acted upon it; that is, before it has described 180°, or is arrived at B. 

In like manner, while the moon is passing from B to A, through F, being drawn toward the plane 
of the ecliptic by the disturbing force, it will cross the ecliptic sooner than it would otherwise have 
done, that is, before it arrives at A. Consequently, the nodes will have changed their places, and 
moved in a contrary direction to the moon. In any other position of the line of the nodes, the dis¬ 
turbing force will, for the same reason, cause the line of the nodes to move in antecedentia , though in 
a less degree; because, whilst the moon is describing the greater part of its orbit, it is drawn by the 
disturbing force (as was shown in the last Prop.) toward the ecliptic, and consequently is made to cross 
the ecliptic sooner than it would otherwise have done, that is, the nodes are on the whole, in one 
revolution of the moon, made to move in a direction contrary to that of the moon ; but this regress 
is less than when the line of the nodes is in quadrature, because, during part of the revolution in this 
oblique position of the line of the nodes, the nodes move in consequentia, or in the same direction with 
the moon, namely, whilst the disturbing force (as was shown in the last Prop.) draws the moon from 
the plane of the ecliptic; whereas, when the line of the nodes is.In quadrature, they move in antece¬ 
dentia during the whole revolution. 

Schol. 1. The nodes perform one revolution, or pass through every part of the ecliptic, in about 
19 years. 

Schol. 2. All the irregularities of the moon are greater when the earth is in its perihelion, than 


I 


OF ASTRONOMY. Book VII. Part II. 

when it is in its aphelion, because the effect of the sun’s action, whereby they are produced, is in¬ 
versely as the cube of its distance from the earth. They are also greater when the moon is in con¬ 
junction with the sun, than in opposition, for the same reason ; for the earth and moon, taken togeth¬ 
er, are nearer the sun in the former situation of the moon, than they are in the latter. 


CHAPTER. III. 

Of the Spheroidical Form of the Earth. 

PROP. CLXVII. In the daily revolution of the earth round its axis, the centrifu¬ 
gal force diminishes the weight of bodies more at the equator than in any other place 
on the surface of the earth, in the duplicate ratio of the semidiameter to the cosine of 
the latitude of the place. 

Plate 11. Let PEP e be the earth, PP the axis, Ee the equator. As the earth revolves upon its axis, every 
Fig. 13. place on its surface, except the two poles, describes a circle, the plane of which is perpendicular to 
the axis, and the radius of which is the distance of that place from the axis. Thus, a body placed at 
A will in one revolution of the earth describe a circle, the semidiameter of which will be AB, which, 
with the plane in which it lies, will be perpendicular to the axis PP In like manner, CE is the semi¬ 
diameter of a circle described by the revolution of a place in the equator. But CE is the semidi¬ 
ameter of the earth, and AB is the cosine of latitude of the place A ; for AB is the sine of AP, the 
complement of AE, which is the latitude of the place. And a body at E, revolving in a circle whose 
radius is CE, performs its revolution in the same time with a body at A, revolving in a circle whose 
radius is AB. But where the periodical times are equal, the centrifugal forces are as the radii, (by 
Book II. Prop. LXXVII.) Whence the body at E has its centrifugal force as much greater than the 
body at A, as the radius CE is greater than the radius AB ; and universally, the centrifugal force at 
the equator is* to the centrifugal force at any other place on the surface of the earth, as the semidiameter 
of the earth to the cosine of the latitude of the place. 

Moreover, if the centrifugal forces at E and A were equal, they would diminish the weights of 
bodies unequally, on account of the different directions 4n which these forces act The centrifugal 
force at A, acting obliquely upon the force of gravitation toward C, can only diminish this force by 
such a part of its action as is opposite to the direction of gravitation, that is, resolving A b which may 
express the centrifugal force at A into A a, a 6, the part of the centrifugal force which will act to 
diminish the gravity of the body at A, will be to the whole centrifugal force at A, as A a to A b. 
Whereas at E, the whole centrifugal force, acting in direct opposition to the force of gravitation, will 
operate to diminish the weight of a body at E. Hence the force which acts to diminish the weight 
of a body, that is, the diminution at E is to the same at A, as the whole centrifugal force A b to the part 
A a. But A b is to A a, as CE to BA ; for the triangles A a b and ABC being similar, A b is to Art, as 
AC or EC to BA. Therefore, from the different directions in which the centrifugal forces act at E 
and A, the weight at E is as much more diminished than at A, as EC, the semidiameter, is greater 
than AB, the cosine of the latitude of the place A. 

The centrifugal force then diminishes the force of gravitation in the ratio of EC to AB, both be¬ 
cause the centripetal force at E is greater than at A, and because it acts directly at E, but obliquely at 
A. Therefore the centrifugal force diminishes the weight of a body at E, more than at A, in the 
duplicate ratio, of CE to BA, that is, as much more at E than at A, as the square of CE is greater than 
the square of BA. 

Schol. It is found by calculation from this Proposition, that gravity at the equator is diminished 
by the centrifugal force in the ratio of 288 to 289. 

Cor. 1. If the diurnal motion of the earth round its axis w r ere about 17 times faster than it is, the 
centrifugal force would, at the equator, be equal to the power of gravity, and all bodies there would 
entirely lose their weight. But if the earth revolved still quicker than this, they would all fly off. 

Cor. 2. Since a place in the equator describes a circle of 24930 miles in 24 hours (Prop. III. 
Cor.) it is evident that the velocity with which that place moves, is at the rate of about 17.3 miles per 
minute. The velocity in any parallel of latitude decreases in the proportion of the cosine of latitude 
to radius. Thus, for the latitude of London, say£ as Rad.: Cos 51° 30 : : velocity of the equator : velocity 
of London ; by logarithms, as 10.00000: 9.A£|Jrt 50 :: 1.232046 : 1.026196 == 10.6 miles ; that is, the city 
of London moves about the axis of the eantmat the rate of more than 10§ miles in a minute of time. 



Chap. IV. 


PRECESSION OF THE EQUINOXES. 21 

PROP. CLX\ III. The earth is an oblate spheroid, elevated at the equator, and 
depressed at the poles. 

It has been found by observation, that a pendulum, shorter by 2.169 lines, is required to vibrate 
seconds at the equator, than at the poles; but (from Book II. Prop. XLI1I. and XLIV.) the lengths of pen¬ 
dulums vibrating in the same time are as the gravities at the places where they vibrate ; therefore the 
gravity at the poles is greater than at the equator. And it has been found by Sir I. Newton, that this dif¬ 
ference of gravity is so much greater than would arise from the centrifugal force alone, that the ratio 
of the equatorial diameter of the earth to the polar diameter, must be as 230 to 229, which makes the 
equatorial diameter exceed the polar by about 34 miles. 

Cor. 1 . Hence bodies near the poles arc heavier than the same bodies toward the equator ; (1.) Be¬ 
cause they are nearer the earth’s centre, where the whole force of the earth’s attraction is accumulated 
(2.) Because their centrifugal force is less on account of their diurnal motion being slower. For both these 
reasons, bodies carried from the poles toward the equator, gradually lose their weight. 

Cor. 2. The degrees of latitude upon the earth’s surface are longer at the poles than at the equator. 

For an arc of a meridian near the poles is less curved than near the equator; that is, it is an arc of a 
larger circle ; w’hence a degree measured upon that arc must be greater than upon an arc of the same 
meridian at the equator. 

Cor. 3. The tendency of a heavy body, on any part of the surface of the earth between the poles 
and the equator, is not directly toward the centre, but toward some point between the centre and the 
equator. 

Schol.- The point toward which a body in any given place will tend may be determined. 

For (by Prop. CLXVII.) as radius EC is to the cosine of latitude, of the place AB, so is the centri- Fkte 11 
fugal force at E to the centrifugal force at A in the direction A b. Produce, therefore, the line BA to b Fi S- 13 ‘ 
till A b has the same ratio to AC, as the quantity last found has to gravity upon the surface of the earth, 
fcomplete the parallelogram Ab C c ; the point sought will be c, and the tendency of the body will be 
along Ac. Thus, suppose the latitude of the place 51° 46'; the centrifugal force at the equator is found 
to be to that of gravity, as 1 to 289; hence, as radius to the cosine of 51° 46', so is 1 to 0.618, which 
is the centrifugal force at A. Consequently, the centrifugal force at A i? to the force of gravity as 0.618 
to 289 ; therefore, by the construction, Ab or C c is to AC in that ratio. The ratio of AC to C c being thus 
found, as AC is to C c, or as 289 is to 0.618, so will the sine of the angle of latitude AC c, or 51° 46', be 
to 5' nearly, which is the angle required, measuring the deviation of the line of direction of falling bodies 
at the given latitude from a line drawn to the centre of the earth. 


CHAPTER IV. 

* Of the Precession of the Equinoxes. 

Def. LXIV. A Periodical Year is the time in which the sun completes its revolu¬ 
tion through the ecliptic. 

Def. LXV. A Tropical Year is the time in which the sun completes its revolution, 
setting out from any solstitial or equinoctial point, and returning to the same. 

PROP. CLXIX. The equinoctial points move in antecedents , or go backward 
from east to west, contrary to the order of the signs. 

It is found from observation, that the equator and ecliptic do not always intersect each other in the 
same points, but that the points of intersection change their place, moving from east to west, whilst the 
inclination of the planes remains the same. This motion is called the precession of the equinoxes, because 
it carries the equinoctial points in preccdentia signa. 

PROP. CLXX. The precession of the equinoxes makes the tropical year shorter 
than the periodical year. 

If, while the sun moves in the order of the signs, the equinoctial point moves in the contrary direc¬ 
tion, it is manifest, that the sun must arrive at the solstitial or equinoctial point from which it set out, 
before it arrives at the same place in the zodiac, or must complete the tropical year sooner than the 
periodical year. 



218 


Plate 11 
Fig. 12. 


OF ASTRONOMY. Rook VII. Part II. 

The tropical year is observed to be 365 days, 5 hours, 49 minutes; the periodical year, 365 days, G 
hours, 9 minutes, 12 seconds. 

PROP. CLXXI. The precession of the equinoxes causes the poles of the equator 
to describe a circle from east to west about the poles of the ecliptic. 

In this precession, the plane of the equator revolves from east to west, cutting the ecliptic, which 
with its axis is at rest, in successive points. But while the plane of the equator is revolving, its axis 
must revolve with it the same way. And since the plane of the equator is always equally inclined to 
that of the ecliptic, the axis of the equator must always have the same inclination to the axis of the 
ecliptic ; consequently, the poles of the equator will revolve round the poles of the ecliptic, always pre¬ 
serving the same distance from each other; that is, the poles of the equator will describe a circle about 
the poles of the ecliptic. 

Exp. The precession of the equinoxes, and the revolution of the pole of the equator about that of 
the ecliptic, may be thus represented on the celestial globe. Let the broad wooden horizon represent 
the ecliptic; place the axis of the globe perpendicular to the wooden circle ; the ecliptic on the globe 
will then make an angle of 23° 30' with the wooden horizon; consequently, if the wooden horizon 
represent the ecliptic, the circle which commonly represents the ecliptic will now represent the 
equator; and the two points in which the circle cuts the wooden horizon will represent the 
equinoctial points. If the globe, in this position, be turned slowly round from east to w r est, these points 
of intersection will move round the same way, while the inclination of the circle which now represents 
the equator to that which represents the ecliptic remains the same; whence the precession of the 
equinoxes is properly represented. Again, the axis and poles of the globe now r representing those of 
the ecliptic, the axis and poles of the ecliptic, marked on the globe, will represent those of the equator; 
and in turning the globe round from east to west, the points which represent the poles of the equator 
will revolve the same way round the poles of the globe which represent those of the ecliptic, and the 
axis of the supposed equator will always make the same angle with the plane of the supposed ecliptic. 

PROP. CLXXII. The precession of the equinoxes is caused by the action of the 
sun and moon on that excess of matter about the equatorial parts of the earth, by which 
from a perfect sphere it becomes an oblate spheroid. 

Let ADCB be the plane of the ecliptic, S the sun, E the earth, and AFBG a ring encompassing the 
earth at any distance, as Satufn is encompassed by its ring. Let the half of this ring AGB toward the 
sun be above the plane of the ecliptic and the other half below it; then, a line passing through A and 
B will be the line of the nodes of this ring. If it be supposed that this ring moves round its centre E, 
the same way in which the moon moves round the earth, it is obvious that every point of this ring will 
be acted upon by the disturbing force of the sun in the same manner as the moon was shown to be act¬ 
ed upon in Prop. CLVllI. &c. Particularly, the motion of the nodes of this ring, and consequently 
of the whole ring which moves with these nodes, and its inclination to the plane in which its centre 
moves, will be affected in the same manner with the orbit of the moon; wdience, its nodes when in 
syzygies will stand still, and its inclination will be greatest; but in all other situations, the nodes will 
go backward, and fastest of all when in the quadratures, at which time the inclination of the ring will 
be the least. This will be the case, whatever be the thickness of the ring, or its distance from the 
centre. 

If this ring be supposed to adhere to the earth, it is obvious that it will still have the motions de¬ 
scribed above, and that, in this situation, the earth itself must participate of these motions. Now the 
earth being an oblate spheroid, having its equatorial diameter longer than that which passes through 
its poles, this redundancy of matter, by which the form of the earth departs from a perfect sphere, 
may be considered as a portion of the supposed ring, which receives from the action of the sun the 
motions abovementioned, and communicates them to the earth. Hence the equinoctial points, which 
are the nodes of the ring, when they are in syzygy, that is, at the equinox, will stand still, and the 
inclination of the equator to the plane of the ecliptic will be the greatest; in all other situations they 
will go backward, and fastest when in quadrature at the solstices ; and the inclination of the plane of 
the equator to that of the ecliptic is then the least. 

Cor. Hence the axis of the earth, being perpendicular to the plane of the equator, changes there¬ 
with its inclination to the plane of the ecliptic tw^ce in every revolution of the earth about the sun. 
For instance, it increases whilst the earth is moving from the solstitial to the equinoctial, and dimin¬ 
ishes as much in its passage from the equinoctial to the solstitial points; which phenomenon is called 
the Nutation of the Poles. , 

Schol. This precession of the equinoxes is found to be 50 seconds of a degree, every year, west¬ 
ward or contrary to the sun’s annual motion; so that, with respect to the fixed stars, the equinoctial 


Chap. V. 


OF THE TIDES. 


219 


points fall backward 30 degrees in 2160 years, whence the stars will appear to have gone 30 degrees 
forward with respect to the signs of the ecliptic, which are reckoned from the equinoctial point. 
Thus the stars which were formerly in Aries are now in Taurus, &c. This period is completed in 
25920 years. _ w 


CHAPTER V. 

Of the Tides. 

PROP. CLXXIII. The titles are caused by the attraction of the moon and of the 
sun. 

Let Ap Lube the earth, and C its centre ; let the dotted circle PN represent a mass of water Plate 11. 
covering the surface of the earth ; let M, to, be the moon ; S, s, the sun, in different situations. Be- Fig- 
cause the power of gravity diminishes as the squares of the distances increase (by Prop. CXLV1II.), the 
waters on the side of the earth A are more attracted by the moon at M, than the central parts of the earth 
C, and the central parts are more attracted than the waters on the opposite side of the earth at L: con¬ 
sequently (as was shown concerning the moon) the waters on the side L will be as it were attracted from. 
the centre of the earth, or will recede from thence. Therefore, while the moon is at M, the waters will 
rise toward a and l on the opposite sides of the earth A, L; while, by the oblique attraction of the moon, 
the waters at P and N will be depressed. 

Or thus ; because (by Prop. CLV.) the moon and earth are continually revolving about their com 
mon centre of gravity, suppose a ; the points A, C, L, describing circles about this common centre in 
the same periodical times, the forces required to retain them in these circles (as may be inferred from 
Book II. Prop. LXXV.) will be to each other as their distances from the centre a A, a C, a L. Conse¬ 
quently, the point L requires a greater force than C, and C than A, to retain it in its orbit. Now these 
points are retained in their respective circles by the moon at M ; and consequently the point L, which 
is most remote, and therefore requires the greatest force, is attracted the least, whilst A, the nearest 
point, is attracted the most. Thus, the water about A being attracted too much, and that about L too 
little, both will have their gravity diminished by the action of the moon, and will endeavour to leave 
the centre C; while the water at P and N, having their gravity increased by the same cause, will 
subside. Hence the form of the water on the surface of the earth will become an oblong spheroid. 

This oval of waters keeps pace with the moon in its monthly course round the earth ; while the 
earth, by its daily rotation about its axis, presents each part of its surface to the direct action of the 
moon, tw'ice each day, and thus produces two floods and two ebbs. But because the moon is in the 
mean time passing from west to east in its orbit, it comes to the meridian of any place later than it 
did the preceding day; whence the two floods and ebbs require nearly 25 hours to complete them. 

The tide is at the greatest height, not when the moon is in the meridian, but some time afterward, be¬ 
cause the force, by which the moon raises the tide, continues to act for some time after it has passed 
the meridian. 

As the moon thus raises the water in one place, and depresses it in another, the sun does the same ; 
but in a much less degree, on account of the small ratio of the semidiameter of the earth to the dis¬ 
tance of the sun ; for, as it was shown of the moon that the force of the sun, by which it disturbs its 
motion, is as the distance of the moon from the earth to that of the sun from the same, so, in this case, 
the force of the sun to disturb the waters is as the semidiameter of the earth, to the distance of the 
sun, which ratio is very small. . * 

Cor. The moon being nearest to the earth when in perigee, its attraction must then be strongest, 
and the effect or elevation of the waters, greatest. And the earth being in its perihelion at the winter 
solstice, the sun’s power to produce tides is greatest at that time. 

PROP. CLXXIV. The tides are greatest at the new and full moons, and least at 
the first and last quadratures; and the Highest of the former tides are near the time 
of the equinoxes. 

When the moon is in conjunction or opposition with the sun, as M, to, S, the tides which each en- p| atell 
deavours to raise are in the same place; whence they arp then greatest, and are called spring Fig. t il 
tides. But, when the moon is in the first or last quarter, the sun, being in the meridian when the moon 
is in the horizon, as M, s , depresses the water where the moon raises it; whence the tides are then 



OF ASTRONOMY. 


Rook VII. Part II. 


2*0 

least, and are called neap tides. On the full and new moons which happen about the equinoxes, when 
the luminaries are both in the equator or near it, the tides are the greatest of all; for, first, the two 
eminences of water are at the greatest distance from the poles, and hence the difference between ebb 
and flood is more sensible ; for if those eminences were at the poles, it is obvious we should not per¬ 
ceive any tide at all; secondly, the equatorial diameter of the earth produced passes through the moon, 
which diameter is longer than any other, and consequently there is greater disproportion between the 
distances of the zenith, centre, and nadir, from the centre of gravity of the earth and moon, in this sit¬ 
uation than in any other; and thirdly, the water rising higher in the open seas, rushes to the shores 
with greater force, where being stopped, it rises higher still; for it not only rises at the shores in pro¬ 
portion to the height it rises to in the open seas, but also according to the velocity with which it flows 
from thence against the shore. The spring tides, which happen a little before the vernal and after 
the autumnal equinox, are the greatest of all, because the sun is nearer the earth in the winter than 
in the summer. 

. f. 

PROP. CLXXV. When the moon is in the northern hemisphere, it produces a 
greater tide while it is in the meridian above the horizon, than when it is in the me¬ 
ridian below it; when in the southern hemisphere, the reverse. 

Plate 11. Let AFHD represent the earth, whose centre is T, and axis PO, the point P the north pole, and 
Fig. 15. q the south pole, EQ the equator, FH a parallel to it on the south side, and KD another parallel to 
it on the north side. Let the fluid surrounding the earth form itself into an oblong spheroid, whose 
longer axis HK produced, passes through the moon at L. The right lines TK, or TPI, drawn from 
the centre T, will represent the greatest height of the water in those places. Then, supposing NM 
perpendicular to KH, TN or TM will denote the least height, and will represent the height of the 
water in all parts of the globe through which the circle NM passes. The right lines TE, TF, TH, 
TQ, TD. will show the height of the water in those respective places E, F, H, Q, D. 

Let us now consider some place in particular, which, by the diurnal motion of the earth, describes 
the parallel KD. When that place is at K, the height of the water TK is the greatest, that is, it will 
be high water, antLthe moon L will be in the meridian. But afterward, when that place comes to X, 
the height of the water will be the least, that is, it will be low w r ater; and when the same place comes 
to D, it will be high w ater again. But because TK is greater than TD, therefore, in the present 
case, w hen the moon is on the north side of the equator, or in the northern signs, the height of the 
sea, or tide, will be greater when the moon is in the part of the meridian, which is above the horizon, 
than when it is in the meridian, and below it. Hence it is that the moon, when it is in the northern 
signs, makes the greatest tides on our side of the equator when it is above the earth. 

Again, TH, on the south side of the equator, is longer than TF; and therefore, to a place that de¬ 
scribes the parallel FH, the greatest height of the water, when the moon is in the northern signs, is 
w’hen it is on that part of the meridian that is below the horizon of that place, and the least tides 
when it is above the horizon For the like reason, when the moon is in the southern signs, the great¬ 
est tides on the other side of the equator will be when it is below our horizon, and the least tides 
when it is above it. 

Cor. Hence it is evident, that when the moon is in the. equator, the two-tides are equally high. 
For then the longer axis I1K of the oblong spheroid coincides with the equator EQ, and the two points 
of high water on any parallel, being equally distant from the highest points of the tides, are conse¬ 
quently at the same height. 

Schol. What has been said of the tides must be understood upon supposition, that the globe of 
of the earth is covered entirely with water to a considerable depth ; but continents which stop the tide, 
straits between them, islands, and the shallowness of the sea in some places, which are all impediments 
to the course of the water, cause many exceptions to what hath been above laid down. These excep¬ 
tions can only be explained from particular observations on the nature of tides at different places. 






\ 




4 


BOOK VII. PART III 


OF THE FIXED STARS. 


Def. LXVII. Those bodies, which always appear in the heavens at the same 
distance from each other, are called Fixed Stars ; because they do not appear to have 
any proper motion of their own. 

PROP. CLXXVI. The fixed stars are luminous bodies. 

Because they appear as points of small magnitude when viewed through a telescope, they must 
be at such immense distances, as to be invisible to the naked eye if they borrowed their light; as is the 
case with the satellites of Jupiter and Saturn, although they appear of very distinguishable magni¬ 
tude through a telescope. Besides, from the weakness of reflected light, there can be no doubt but 
that the fixed stars shine with their own light. They are easily known from the planets, by their 

twinkling. . ^ 

Schol. The number of stars, visible at any one time to the naked eye, is about 1000; but Dr. Her- 
schel, by his magnificent improvements of the reflecting telescope, has discovered that the whole num¬ 
ber is great beyond all conception. 

PROP. CLXXVII. The fixed stars appear of different magnitudes. 

The magnitudes of the fixed stars appear to be different from one another, which difference may 
arise either from a diversity in their real magnitudes, or distances; or from both these causes acting 
conjointly. The difference in the apparent magnitude of the stars is such as to admit of their being 
divided into six classes, the largest being called stars of the first magnitude, and the least, which are 
visible to the naked eye, stars of the sixth magnitude. - Stars only visible by the help of glasses arc 
called telescopic stars. 

Schol. 1. It must not be inferred that all the stars of, each class appear exactly of the same mag¬ 
nitude • there being great latitude given in this respect ; even those of the first magnitude appear 
almost all different in lustre and size. There are also other stars of intermediate magnitudes, which as 
astronomers cannot refer to any one class, they therefore place them between two. Procyon , for in¬ 
stance which Ptolemy makes of the first magnitude, and Tycho of the second, Flamstead lays down 
as between the first and second. So that instead of 6 magnitudes, we may say that there are almost 
as many orders of stars, as there are stars ; such considerable varieties being observable in their 

magnitude, colour, and brightness. _ 

Schol. 2. To the bare eye the stars appear of some sensible magnitude, owing to the glare of light 
arising from the numberless reflections of the rays in coming to the eye ; this leads us to imagine that 
the stars are much larger than they would appear, if we saw them only by the few rays which come, 
directly from them, so^as to enter the eye without being intermixed with others. 

Exr Examine a fixed star of the first magnitude through a long and narrow tube ; which, though it 
takes in as much of the sky as w ould hold a thousand such stars, scarcely renders that one visible. 

Schol. 3. There seems but little reason to expect that the real magnitudes of the fixed stars will ever 
be discovered with certainty; we must, therefore, be contented with an approximation, deduced from 
their parallax (if this should ever be ascertained), and the quantity of light they afford us compared 
with the sun. To this purpose, Dr. Herschel informs us that with a magnifying power of 6450, and by 
meAns of his new micrometer, he found the apparent diameter of * Lyrae to be 0.335". 

Dr HerschePs method of finding the annual parallax of the fixed stars is by observing how the angle 
betw een two stars very near to each other varies in opposite parts of the year. The following is the 
most simple case given by that great astronomer. Let G and E be two stars situated in a line with the Pla(e n 
earth at and supposed perpendicular to the diameter AB of the earth’s orbit, and when the earth is*fig. 16. 
at b observe the angle GBE. Let P= the angle AGB, or the annual parallax ofG; p the angle GBE 
found by observation'; M, m, the angles under which the diameters of G and E appear, and draw GFI 
perpendicular to BG. Then p : P :: GH 

_ p X M the para ii ax g. If G be a star of the first magnitude, and 
M — rn 
then P = li"- 


AB :: GE : AE :: (because M : m: : AE : AG) M —m : M ; hence 

E one of the third, and p — 1", 


See Phil. Trans. Vol. lxxii. 




222 


OF ASTRONOMY. 


Book VII. Past. II. 


This theory is only true upon the supposition that the stars are all of the same magnitude ; and that 
a star of the second magnitude is at double the distance of one of the first, and so on. These supposi¬ 
tions are certainly not founded on any analogy from the known and well established principles of that 
system of bodies to which we belong. 

Schol. 4. The ingenious observation of Kepler upon the magnitudes and distances of the fixed stars 
deserves to be introduced here, and the more so, as he was followed in the conjecture by the great Dr. 
Halley. He observes, that there can be only 13 points upon the surface of a sphere as far distant from 
each other as from the centre; and supposing the nearest fixed stars to be as far from each other as 
from the sun, he concludes that there can be only 13 stars of the first magnitude. Hence at twice that dis¬ 
tance from the sun there may be placed four times as many, or 52. At three times that distance, nine 
times as many, or 117; and so on. These numbers will give pretty nearly the number of stars of the 
first, second, third, &c. magnitudes. Dr. Halley farther remarks, that if the number of stars be finite, 
and occupy only a part of space, the outward stars would be continually attracted to those within, and 
in time would unite into one. But if the number be infinite, and they occupy an infinite space, all the 
parts w’ould be nearly in equilibrio, and consequently each fixed star being drawn in opposite directions 
would keep its place, or move on till it had founed an equilibrium. Phil. Trans. No. 364. See also 
the introductory remarks of Dr. Herschel to a paper on the changes of the fixed stars. Phil. Trans. 
1796. 

PROP. CLXXVIII. The fixed stars are divided into constellations, or systems of 
stars. 

The ancients, that they might the better distinguish the stars with regard to their situation in the 
heavens, divided them into several constellations, that is, systems of stars, each system consisting of 
such as are near each other. And to distinguish these systems from one another, they gave them the 
names of such men or things as they fancied the space they took up in the heavens represented. To 
these, several new constellations have been added by modern astronomers. 

Schol. The following table contains the names of the constellations, and the number of stars observ¬ 
ed in each by different astronomers. 


The Ancient Constellations. 


Ptolemy. 

Tycho. 

Hevelius. 

Flamstead. 

Ursa Minor 

The Little Bear 

8 

7 

12 

24 

Ursa Major 

The Great Bear 

35 

29 

73 

87 

Draco 

The Dragon 

31 

32 

40 

80 

Cepheus 

Cepheus 

13 

4 

51 

35 

Bootes, Arctopliilax 

Bootes 

23 

18 

52 

54 

Corona Borealis 

The Northern Crown 

8 

8 

8 

21 

Hercules, Engonasin 

Hercules kneeling 

28 

28 

45 

113 

Lyra 

The Harp 

10 

11 

17 

21 

Cygnus, Gallina 

The Swan 

19 

18 

47 

81 

Cassiopea 

The Lady in her Chair 

13 

26 

37 

55 

Perseus 

Perseus 

29 

29 

46 

59 

Auriga 

The Wagoner 

14 

9 

40 

66 

Serpentarius, Ophiuchns 

Serpentarius 

29 

15 

40 

74 

Serpens 

The Serpent 

18 

13 

22 

64 

Sagitta 

The Arrow 

5 

5 

5 

18 

Aquila, Vultur 

The Eagle ) 


12 

23 


Antinous 

Antinous $ 

15 

3 

19 

71 

Delphinus 

The Dolphin 

10 

10 

14 

18 

Equulus, Equi sectio 

The Horse’s Head 

4 

4 

6 

10 

Pegasus, Equus 

The Flying Horse 

20 

19 

38 

89 

Andromeda 

Andromeda 

23 

23 

47 

66 

Triangulum 

The Triangle 

4 

4 

12 

16 

Aries 

The Ram 

18 

21 

27 

66 

Taurus 

The Bull 

44 

43 

51 

141 

Gemini 

The Twins 

25 

25 

38 

85 

Cancer 

The Crab 

23 

15 

29 

83 

Leo 

The Lion > 


30 

49 

95 

Coma Berenice? 

Berenice’s Hair $ 

35 

14 

21 

43 


Book VII. Part III. OF THE FIXED STARS. 


The Ancient Constellations. 

Ptolemy. 

Tycho. 

Hevcliui. 

Flamstead, 

Virgo 

The Virgin 

32 

33 

50 

110 

Libra, Chelte 

The Scales 

17 

10 

20 

51 

Scorpius 

The Scorpion 

24 

10 

20 

44 

Sagittarius 

The Archer 

31 

14 

22 

69 

Capricornus 

The Goat 

28 

28 

29 

51 

Aquarius 

The Water-bearer 

45 

41 

47 

108 

Pisces 

The Fishes 

38 

36 

39 

113 

Cetus 

The Whale 

22 

21 

45 

97 

Orion 

Orion 

38 

42 

62 

78 

Eridanus, Flavius 

Eridanus, the River 

34 

10 

27 

84 

Lepus 

The Hare 

12 

13 

16 

19 

Canis major 

The Great Dog 

29 

13 

21 

31 

Canis minor 

The Little Dog 

o 

2 

13 

14 

Argo Navis 

The Ship 

45 

3 

4 

64 

Hydra 

The Hydra 

27 

19 

31 

CO 

Crater 

The Cup 

7 

3 

10 

31 

Corvus 

The Crow 

7 

4 


9 

Centaurus 

The Centaur 

37 



35 

Lupus 

The Wolf 

19 



24 

Ara 

The Altar 

7 



9 

Corona Australis 

The Southern Crown 

13 



12 

Piscis Australis 

The Southern Fish 

18 



24 

The New 

Southern Constellations, 





Columba Noachi 

Noah’s Dove 



10 


Robur Carolinum 

The Royal Oak 



12 


Grus 

The Crane 



13 


Phoenix 

The Phenix 



13 


Indus 

The Indian 



12 


Pavo 

The Peacock 



14 


v Apus, Avis Indica 

The Bird of Paradise 



11 


Apis, Musca 

The Bee, or Fly 



4 


Chamaeleon 

The Camelion 



10 


Triangulum Australe 

The South Triangle 



5 


Piscis volans, Passer 

The Flying Fish 



8 


Dorado, Xiphias 

The Sword Fish 



6 


Toucan 

The American Goose 



9 


Hydrus 

The Water Snake 



10 


Hevelius' , s Constellations made out of the unformed Stars. 






Hevelius. 

Flamsteati. 

Lynx 

The Lynx 


19 

44 


Leo minor 

The Little Lion 



53 


Asterion and Chara 

The Grey Hounds 


23 

25 


Cerberus 

Cerberus 


4 



Vulpecula & Anser 

The Fox and Goose 


27 

35 


Scutum Sobieski 

Sobieski’s Shield 


7 



Lacerta 

The Lizard 


10 

16 


Camelopardalis 

The Camelopard 


32 

58 


Monoceros 

The Unicorn 


19 

31 


Sextans 

The Sextant 


11 

41 



Schol. 1 . Stars not included in any constellation are called unformed stars. Besides the names o 
the constellations, the ancient Greeks gave particular names to some single stars, or small collection 
of stars ; thus, the cluster of small stars in the neck of the bull, was called the Pleiades ; five stars in 
the Bull’s face, the Hyades ; a bright star in the breast of Leo, the Lion's Heart; and a large star be¬ 
tween the knees of Bootes , Arcturus. 

Schol. 2. The constellations may be represented on two plane spheres projected on a great circle, 
or on the convex surface of a solid sphere, as on the celestial globe, or most perfectly on the concave 
surface of a hollow sphere. If the celestial globe be made use of, after rectifying it to the time of the 
night, the stars may be found, by conceiving a line drawn from the centre of the globe through any 
star in the heavens, and its representation upon the globe. Greek letters have been added by Bayer | 


223 


/ 


224 “ OF ASTRONOMY. Rook VII. Part III. 

to stars in the several constellations of his catalogue (* being affixed to the largest star), by means of which 
any star may be easily found. 

Schol. 3. Twelve of these constellations lie upon the ecliptic, including a space about 15° broad, 
called the Zodiac , within which all the planets move. The constellation Aries, about 2000 yejrs ago, 
lay in the first sign of the ecliptic ; but on account of the precession of the equinoxes, it now lies in the 
second. Prop. CLXXII. Schol. In the foregoing table Antinous was made out of the unformed stars near 
Aquila; and Coma Berenices out of the unformed stars near the Lion’s tail. They are both mentioned 
by Ptolemy, but as unformed stars. The constellations as far as the triangle, with Coma Berenices , 
are northern ; those after Pisces are southern. 

PROP. CLXXIX. The luminous part of the heavens, called the Milky Way , con¬ 
sists of fixed stars too small to be seen by the naked eye. 

This is found from observations made with telescopes. 

In a paper on the Constructions of the Heavens, Dr. Herschel says, {t it is very probable, that the 
great stratum called the milky way is that in which the sun is placed, though perhaps not in the cen¬ 
tre of its thickness, but not far from the place where some smaller stratum branches from it. Such a 
supposition will satisfactorily, and with great simplicity, account for all the phenomena of the milky 
way, which, according to this hypothesis, is no other than the appearance of the projection of the stars 
contained in this stratum, and its secondary branch.” 

In another paper on the same subject, he says, “ that the milky way is a most extensive stratum of 
stars of various sizes, admits no longer of the least doubt; and that our sun is actually one of the 
heavenly bodies belonging to it is as evident.” 

w We will now retreat to our owqJ retired station in one of the planets attending a star in the 
great combination with numberless others; and in order to investigate what will be the appearances 
from this contracted situation, let us begin with the naked eye. The stars of the first magnitude be¬ 
ing in all probability the nearest, will furnish us with a step to begin our scale ; setting off, therefore, 
with the distance of Sirius or Arcturus, for instance, as unity, we will at present suppose, that those 
of the second magnitude are at double, and those of the third at treble the distance, and so forth. 
Taking it, then, for granted, that a star of the seventh magnitude is about seven times as far from us 
as one of the first, it follows that an observer, who is enchrfd in a globular cluster of stars, and not far 
from the centre, will never be able, with the naked eye, In see to the end of it. For since, according 
to the above estimations, he can only extend his view about seven times the distance of Sirius, it cannot 
he expected that his eyes should reach the borders of a cluster, which has, perhaps, not less than fif¬ 
ty stars in depth every where around him. The whole universe, therefore, to him, will be comprised 
in a set of constellations, richly ornamented with scattered stars of all sizes. Or, if the united bright¬ 
ness of a neighbouring cluster of stars should, in a remarkably clear night, reach his sight, it will put 
on the appearance of a small, faint, nebulous cloud, not to be perceived without the greatest attention. 
Allowing him the use of a common telescope, he begins to suspect that ail the milkiness of the bright 
path which surrounds the sphere may be owing to stars. By increasing his power of vision, he be¬ 
comes certain, that the milky way is indeed no other than a collection of very small stars, and the 
nebulae nothing but clusters of stars.” 

Dr. Herschel then solves a general problem for computing the length of the visual ray; that of the 
telescope, which he uses, will reach to stars 497 times the distance of Sirius. Now (by Prop. B. Cor. 
1. p. 323.) Sirius cannot be nearer than 100,000 x 190,000,000 miles; therefore Dr. Herschel’s teles¬ 
cope will, at least, reach to 100,000 x 190,000,000 x 497 miles. And Dr. Herschel says, that in the 
most crowded part of the milky way, he has had fields of view that contained no less than 580 stars, and 
these were continued for many minutes, so that, in a quarter of an hour, he has seen 116,000 stars 
pass through the field view of a telescope of only 15' aperture; and at another time, in 41 minutes, 
he saw 258,000 stars pass through the.field of his telescope. Every improvement in his telescopes 
has discovered stars not seen before, so that there appear no bounds to their number, nor to the ex¬ 
tent of the universe. See Phil. Trans. Vol. lxxiv^and lxxvi. 

Schol. 1. There are spots in the heavens, called Nebulce , some of which consist of clusters of teles¬ 
copic stars, others appear as luminous spots of different forms. The most considerable is one in the 
mid-way between the two stars on the blade of Orion’s sword, marked 9 by Bayer, discovered in the 
year 1656 by Huygens; it contains only seven stars, and the other part is a bright spot upon a dark 
ground, and appears like an opening into brighter regions beyond, Dr. Halley and others have dis¬ 
covered nebulae in different parts of the heavens. In the Connoissance des Temps for 1783 and 1784, 
there is a catalogue of 103 nebulae observed by Messier and Mechain. But to Dr. Herschel we are 


I 


Book. VII. Part III. 


OF THE FIXED STARS. 


325' 


indebted for catalogues of 2000 nebulae and clusters of stars, which he himself has discovered. Some 
of them form a round compact system, others are more irregular, of various forms, and some are long 
and narrow. The globular systems ot stars appear thicker in the middle, than they would do if the 
stars were all at equal distances from each other; they are therefore condensed toward the centre. 
That stars should be thus accidentally disposed is too improbable a supposition to be admitted ; he 
supposes, therefore, that they are brought together by their mutual attractions, and that the gradual 
condensation toward the centre is a proof of a central power of such a kind. He observes also, that 
there are some additional circumstances in the appearance of extended clusters and nebulae, that very 
much favour the idea of a power lodged in the brightest part. For, although the form of them be not 
globular, it is plain that there is a tendency to sphericity. As the stars in the same nebulae must be 
very nearly all at the same relative distances from us, and they appear nearly of the same size, their 
real magnitudes must be nearly equal. Granting, therefore, that these nebulae and clusters of stars 
are formed by mutual attraction, Dr. Herschel concludes that we may judge of their relative age by 
the disposition of their component parts, those being the oldest which are most compressed. He sup¬ 
poses, and indeed offers powerful arguments to prove, that the milky way is the nebulae, of which our 
sun is one of the component parts. See Phil. Trans, Vol. lxxvi. and lxxix. 

Schol. 2. Dr. Herschel has also discovered other phenomena in the heavens which he calls nebulous 
stars; that is, stars surrounded with a faint luminous atmosphere of large extent. Those which have 
been thus styled by other astronomers, he says, ought not to have been so called, for on examination 
they have proved to be either mere clusters of stars plainly to be distinguished by his large telescopes, 
or such nebulous appearances as might be occasioned by a multitude of stars at a vast distance. The 
milky way consists entirely of stars ; and he says, “ 1 have been led on by degrees from the most evi¬ 
dent congeries of stars to other groups in which the lucid points were smaller, but still very plainly 
to be seen; and from them to such wherein they could but barely be suspected, until I arrived at last 
to spots in which no trace of a star was to be discerned. But then the gradation to these latter was 
by such connected steps, as left no room for doubt but that all these phenomena were equally occasioned 
by stars variously dispersed in the immense expanse of the universe.” 

In the same paper is given an account of some nebulous stars, one of which is thus described. 
“Nov. 13, 1790. A most singular phenomenon! a star of the eighth magnitude, with a faint lumin¬ 
ous atmosphere of a circular form, and about 3' in diameter. The star is perfectly in the centre, and 
the atmosphere is so diluted, faint, and equal throughout, that there can be no surmise of its consisting 
of stars, nor can there be a doubt of the evident connexion between the atmosphere and the star. An¬ 
other star, not much less in brightness, and in the same field of view with the above, was perfectly 
free from any such appearance.” Hence Dr. Herschel draws the following consequences: Granting 
the connexion between the star and the surrounding nebulosity, if it consist of stars very remote which 
give the nebulous appearance, the central star which is visible, must be immensely greater than the 
rest; or if the central star be no bigger than common, how extremely small and compressed must be + 
those other luminous points which occasion the nebulosity ! As, by the former supposition, the luminous 
central point must far exceed the standard of what we call a star ; so in the latter, the shining matter about 
the centre will be too small to come under the same denomination; we, therefore, either have a central 
body which is not a star, or a star which is involved in a shining fluid, of a nature totally unknown to 
us. This last opinion Dr. Herschel adopts. Light reflected from the star could not be seen at this dis¬ 
tance. Besides, the outward parts are nearly as bright as those near the star. Moreover, a cluster 
of stars will not so completely account for the milkiness or soft tint of the light of these nebulae, as 
a self-luminous fluid. “ What a field of novelty,” says Dr. Herschel, “is here opened to our concep¬ 
tions ! A shining fluid, of a brightness sufficient to reach us from the regions of a star of the 8th, 9th, 

10th, 11th, 12th magnitude, and of an extent so considerable as to take up 3, 4,5, or 6 minutes in diame¬ 
ter. He conjectures that this shining fluid may be composed of the light perpetually emitted from 
millions of stars. See Phil. Trans. Vol. Ixxxi. p. 1, on Nebulous Stars properly so called. 

Schoi.. 3. New stars sometimes appear while others disappear. Several stars, mentioned by an¬ 
cient astronomers are not now to be found ; several are now visible to the naked eye, which are not mem 
tioned in the ancient catalogues; and some stars have suddenly appeared, and again, after a conside¬ 
rable interval, vanished; also a change of place has been observed in some stars. 

The following are remarkable and well authenticated examples. The first new star we have an 
accurate account of, is that discovered by Cornelius Gemma, on Nov. 8, 1572, in the Chair of Cassiopeia. 

It exceeds Sirius in brightness, and Jupiter in apparent magnitude. Tycho Brahe observed it, and found 
that it had no sensible parallax It gradually decayed, and, after 16 months, disappeared. 

On August 13, 1596, David Fabricius observed a new star in the Neck of the Whale , 25° 45' of Aries, 
with 15° 54' south latitude. It disappeared after October in the same year; was discovered again in 
1637. 


29 


226 


OF ASTRONOMY. 


Book VII. Tart III. 


In the year 1600, William Jansenus discovered a changeable star in the neck of the Swan. It was 
seen by Kepler, who wrote a treatise upon it, and determined its place to be 16° 18' ££ , and 55° 30' or 
32' north latitude. Ricciolus saw it in 1616, 1621, and 1624, and is certain that it was invisible from 
1640 to 1650. M. Cassini saw it again in 1655 ; it increased till 1660; then decreased ; and at the end 
of 1661, it disappeared In November, 1665, it appeared again, and disappeared in 1681. In 1715 
it appeared, as it does at present, of the 6th magnitude. 

In 1686 Kircher observed % in the Swan , to be a changeable star; and from 20 years 1 observations, 
the period of the return of the same phases was found to be 405 days; the variations of its magnitude 
are subject to some irregularity. 

In the year 1604, Kepler discovered a new star near the heel of Serpentarins , so very brilliant, that 
it exceeded every fixed star, and even Jupiter, in apparent magnitude. 

Montanari discovered two stars in the Ship, marked j8 and y by Bayer, to be wanting. He saw 
them in 1664, but lost them in 1668. He observed also that fi, in Medusa's Head , varied in its magni¬ 
tude. x 

Mr. Goodricke has determined the periodical variation of Algol, or /3, in Medusa's Head (observed by 
Montanari to be variable), to be about 2 days 21 hours. Its greatest brightness is of the second magni¬ 
tude, and least of the fourth. Phil. Trans. Vol. Ixxiii. 

Schol. 4. From an attentive examination of the stars with good telescopes, many, which appear 
only single to the naked eye, are found to consist of two, three, or more stars. Dr. Maskelyne had ob¬ 
served u Herculis, to be a double star. Dr. Hornsby found tt Bootis to be double. Other astronomers 
had made similar discoveries. But Dr. Herschel, by his highly improved telescopes has found aboujt 
700, of which not more than 42 had been observed before. 

The following are a few of the most remarkable; 

« Herculis, Flam. 64, a beautiful double star; the two stars very unequal; the largest red, and the 
smallest blue, inclining to green. 

u Geminorum, Flam. 66, double, a little unequal, both white; with a power of 146 their distance 
appears equal to the diameter of the smallest. 

c Lyra, Flam. 4 and 5, a double-double star; at first sight it appears double at a considerable dis¬ 
tance, and by a little attention each will appear double ; one set are equal, and both white ; the other 
unequal, the largest white, and the smallest inclined to red. 

/3 Lyra, Flam. 10, quadruple, unequal, white; but three of them a little inclined to red. 

A Orionis, Flam. 39, quadruple, or rather a double star, and has two more at a small distance; the 
double star considerably unequal, the largest white ; smallest, pale rose-colour. 

f*. Herculis, Flam. 86, double, ver} r unequal; the small star is not visible with a power of 278, but is 
seen very well with one of 460; the largest is inclined to a pale red ; smallest, duskish. 

u Lyra, Flam. 3, double, very unequal, the largest a fine, brilliant white, the smallest dusky; it ap¬ 
pears with a power of 227. Dr. Herschel measured the diameter of this fine star, and found it to be 
0''.3553. 

The examination of double stars with a telescope is a very excellent and ready method of proving 
its powers. Dr. Herschel recommends the following method. The telescope, and the observer, having 
been some time in the open air, adjust the focus of the telescope to some single star of nearly the same 
magnitude, altitude, and color of the star, to be examined ; attend to all the phenomena of the adjusting 
star as it passes through the field of view ; whether it be perfectly round, and well defined, or affected 
with little appendages playing about the edge, or any other circumstances of the like kind. Such de¬ 
ceptions may be detected by turning the object-glass a little in its cell, when these appendages will turn 
the same way. Thus you may detect the imperfections of the instrument, and therefore will not be 
deceived when you come to examine the double star. Phil. Trans. Vol. lxxii. and lxxv. 

Schol. 5. The number of stars is unknown. The catalogue published by Bayer contains 1160; that 
by Flamstead, which includes many telescopic stars, contains 3000. But the most complete catalogue is 
that published by the Rev. Mr. Wollaston, in 1789. 

PROP. CLXXX. The longitude of the fixed stars increases, while their latitude 
remains the same. 

Because the vernal equinoctial point (by Prop. CLXXI.) moves westward, the distance between any 
given star and that point, that is, its longitude, will increase. But since this change is produced by the 
precession of the equinoxes, which is performed round the axis of the ecliptic, this motion will make 
no change in the distance of the fixed star from the ecliptic, that is, in its latitude. 

Cor. Hence the constellations of the zodiac are to the east of those signs or arcs of the zodiac which 


Book VII. Part III. 


OF THE FIXED STARS. 


227 




are called by the same names. The first part of the constellation Aries, by the precession of the equi¬ 
noxes, has gone so far to the east, since the names were first given to the signs, that it is now 30° from 
the first degree of Aries in the line of the ecliptic. 

Def. LXVIII. The Annual Parallax of a heavenly body is the change of its appar¬ 
ent place, as it is viewed from the earth in its annual motion. 

If ADBC be the orbit of the earth, S the sun, and A, B, the earth in opposite parts of its orbit; the Plate 11. 
change in the apparent place of any bod}', as viewed from A and from B, is its annual parallax. fig- I 6 -' 

PROP. CLXXXI. The annual parallax of any heavenly body is proportional to the 
angle which a diameter of the earth's orbit would subtend, if it were viewed from that 
body. 

If when the earth is at A, the fixed star E appears at or near the pole, and when the earth has pass¬ 
ed to the opposite point B, a different star F appears at or near the same pole, the star E will have 
changed its place in respect of the pole; for when the pole is at F, the star E, which was at or near it 
before, is at the distance EF from it; the apparent length of this distance EF (by Def. LXVIII.) is the 
star's annual parallax. Now, if AB, the diameter of the earth’s orbit, were to be viewed from the star 
E, it would subtend the angle AEB; but because the axis of the earth is always parallel to itself, AE 
and BF, which coincide with the axis, are likewise parallel ; whence (El. I. 29.) the angle EBF, subtend¬ 
ed by EF, is equal to AEB, subtended by AB ; and AEB is the parallactic angle. 

Cor. The annual parallax of any heavenly body is inversely as its distance from the earth ; for the 
angle AEB (by Book VI. Prop. LXIX.) is inversely as the distance of AB, the axis of the earth’s orbit. 

PROP. A. If the distance of an object be greater than 400,000 times the base, the 
angles at the stations will not sensibly differ from right angles; consequently the lines 
drawn from the object to the station are, physically speaking, parallel. 

Suppose one of the angles to be 90°. Then, since the most accurate instruments for the mensuration 
of angles cannot be depended upon to less than 0'\5, the tangent of which is to radius as 1 to 403,132, if 
the base have a le^s ratio to the distance than this, that is, than about 1 to 400,000 (for the angle, the 
tangent of which is to radius in this ratio is 2".06, or very little more than two seconds), the angle at 
the other station will not sensibly differ from 90°. See Hutton’s Logarithms. 

Schol. It has been seen that Dr. Herschel depends upon his instruments for the accuracy of measur¬ 
ing quantities much less than 0".5. 

PROP. B. The parallax of an object, the distance of which is above 400,000 times 
greater than that between the two stations of observation, is insensible. 

If the object be at a greater distance from either station than 400,000 times the base, the angle at 
one of the stations being 90°, the angle at the other will be more than 89° 59' 57.9", the difference of 
which angle and 90°, being scarcely more than 0' .5, is too small to become sensible by observation. 

Cor. 1 . If the parallax of an object (observed with an instrument sufficiently exact to measure an 
angle of 0."5) De insensible, the distance of it from either station cannot be less than 100,000 times the 
base from the extremities of which it is observed. 

Schol. It is to be remarked, that, though the distance of the object cannot be less than 100,000 times 
the base, yet it may be greater in any assignable ratio. 

Cor. 2. Lines drawn from any given points in a base, to an object, may be esteemed, in practice, 
parallel, without any sensible error, if the distance of the object be more than 100,000 times the base. 

Cor. 3. Rays, therefore, diverging from any point in the sun’s disk upon the surface of the earth, 
may be esteemed parallel, if their distance from each other do not exceed about 970 miles at the earth’s 
surface ; because 970 is to the distance of the earth from the sun in a proportion of 1 to 400,000. 

PROP. CLXXXII. The fixed stars have no sensible annual parallax. 

When the place of the star E is observed by the best instruments from opposite points of the earth’s Plate 11. 
orbit, its apparent place in the heavens remains the same, which could not be the case if the angle of Fi g- lf >- 
its parallax w'ere so much as tw'O seconds. 

Cor. 1. Hence it appears, that the fixed stars are so remote, that a diameter of the earth’s orbit, 
bears no proportion to their distance, or (by Prop. CLXXXI.) that a diameter of the earth’s orbit, it 
viewxd from one of the fixed stars, would appear as a point. 


228 


OF ASTRONOMY. 


Book VII. Part III. 


Plate 11. 
Fig. 17. 


Cor. 2. The distance of the stars must be greater than 400,000 times the base, from the extremities 
of which it is observed ; that is, greater than 400,000 times the diameter of the orbit of the earth, or 
greater than 400,000 X 190,000,000. 

Cor. 3. Two planes drawn parallel to each other, and passing through the extremities of a diame¬ 
ter of the orbit of the earth, if produced, will appear to coincide with the same great circle of the 
heavens ; because the diameter of the earth’s orbit, when seen from the fixed stars, subtends an angle 
less than In the same manner, if a plane passing through the earth’s centre, be parallel to a plane 
drawn to the surface, these planes, when produced, apparently coincide with the same great circle in 
the heavens. 

Cor. 4. The parallax of a fixed star, being not more than0".5, the sun, when viewed from that star, 

32' 6'' 1" 

would appear under an angle less than —--or less than-, and therefore could not be disfinguish- 

11 a 200,U00 100 ’ ° 

ed from a point. 

Sciiol. Since bodies equal in magnitude and splendour to the sun, being placed at the distance of 
the fixed stars, would appear to us as the fixed stars now do, it may be supposed probable, that the fixed 
stars are bodies similar to the sun, which is the centre of the solar system. This being the case, the 
reason will appear, why a fixed star, when viewed through a telescope magnifying 200 times, appears 
no other than a point. For the apparent diameter of the star being less than T ^ part of a second when 
magnified 200 times, will subtend an angle less than 0".5, at the eye of a spectator, observing it in the 
telescope. 

The parallax of the fixed star, w'hen viewed from the opposite parts of the earth’s orbit, is here 
assumed 0".5, but it is probable that the parallax of the nearest star is much less, and consequently the 
distance greater, in the same proportion, as the parallax is less. 

PROP. CLXXXIII. The motion of the earth, and the progressive motion of light, 
will make a fixed star, which has no sensible parallax, deviate from its true place in 
the direction in which the earth moves. 

If a star passes through the zenith of any place when the earth is at A, it will (by last Prop.) pass 
through the zenith of the same place when the earth is at B, the opposite extremity of the earth’s orbit. 
Consequently, such a star might be seen through a verti. al telescope in the same perpendicular at any 
point of the earth’s orbit, if the motion of light from the star were instantaneous. But the progressive 
motion of light will cause the star to deviate from the perpendicular; for, let the earth be moving from 
B to A, and let the velocity of light be to the velocity of the earth, as CA to BA, and let CB be the 
diagonal of the parallelogram formed from CA, BA. Then the direction of a telescope, in order to see 
the star S when the earth is arrived at A, must be AH, parallel to BC. For, suppose BC to be a very 
long, slender telescope, through which only one ray of light could pass at a time, or to be the axis of 
a larger telescope. The star S cannot be seen through this telescope, but through a telescope perpen¬ 
dicular to B, if the earth be stationary at B, and the progress of light instantaneous. But if the tele¬ 
scope in the position BC were to continue in this position, and to move along with the earth to A, so as 
to come into the situation AH, when the earth arrives at A, the star S might then be seen through it. 
For, since the straight course of the ray is the line CA, in which it must always be if it comes to the 
eye without interruption; and since the ray cannot come directly along CB, the axis of the telescope, 
and arrive at the eye in this axis, unless it is always in the axis; that is, since the ray, in order to 
come to the eye, must be always in the line CA, and also in the line CB, it must be always in the com¬ 
mon intersection of these two lines. KowC is the common intersection when the earth is at B; e is the 
common intersection when the earth is at E ; J] when it is at F; g, when it is at G; and A when at A ; 
the telescope, at each station, being successively in the situations CB, EE, FF, GG, HA. Thus the com¬ 
mon intersection descends down the line CA, while the earth moves from B to A; and since the veloci¬ 
ty of light is to that of the earth, as CA to BA, a ray of light will likewise have descended down CA, 
while the earth is moving from B to A. Therefore, in the whole motion of the telescope, the ray will 
have beenin the common intersection in the line CA, and the axis of the telescope, and consequently will 
have passed along the axis of the telescope, and will come without interruption to the eye at A. 

Thus it appears, that by the progressive motion of light, a ray which, coming from S, enters, at C, a 
telescope in the situation CB, will arrive at the eye, when the telescope, carried along BA with a 
velocity which is to that of a ray of light, as BA to CA, is come into the situation HA; and conse¬ 
quently (Book VI. Prop. II.) the eye will see the star through the telescope in the direction AH, the 
axis of the telescope ; that is, some point in the line AH produced will be the apparent place of the 
star. Thus the star’s apparent place has deviated from its true place S in the direction BA, in which 
the earth was moving, so that, if the motion of the earth is from north to south, the star, which ap- 




Book VII. Part III. 


OF THE FIXED STARS. 


229 


peared in the zenith of the place when the earth was at B, will appear to the southward of the ze¬ 
nith when the earth is arrived at A, and the reverse when the earth is moving from south to north. 

According to Bradley’s observations, made on the star y in the constellation Dragon, this star devia¬ 
ted southward from the zenith from Dec. till March, when it had departed from the zenith 20''. From 
that time till June its southern deviation deci'eased, after which it deviated northward, and in Septem¬ 
ber appeared about 20" toward the north of its station in June, from which time till December it con¬ 
tinued returning to its first situation. Thus the deviation of the star was always in the direction of 
the earth’s motion, and contrary to that of any deviation which might be supposed to arise from the 
annual parallax of the star. But such a deviation could not happen unless the earth moved, and the 
motion of light was progressive ; for if the earth did not move, since the star is fixed, no alteration 
could be made in the apparent place of the star by the progressive motion of its rays in a vertical di¬ 
rection ; and if the earth moves, and the propagation, of light were instantaneous, the earth’s velocity 
would be nothing in respect of the velocity of light, or BA with respect to CA would be nothing; 
whence the angle ACB, and its alternate angle CAH, would vanish, and AH wonld become coincident 
with AC, and consequently the star would have no deviation from its true place. Hence we may con¬ 
clude from the deviation of the star above described, both that the earth moves, and that the motion 
of light is progressive. 

Cor. From these observations it is found, that the velocity of star-light is such as carries it through 
a space equal to the sun’s distance from the earth in 8' 13''. 

Schol. Sir Isaac Newton has shown that the sun, by its attractive power, retains the planets belonging 
to our system in their orbits; he has likewise pointed out the method whereby the quantity of matter 
contained in the sun may be accurately determined. Dr. Bradley has assigned the velocity of the solar 
light with a degree of precision exceeding our utmost expectation. Galileo and others have ascer¬ 
tained the rotation of the sun upon its axis, and determined the position of its equator. By means of 
the transit of Venus over the disk of the sun, our mathematicians have calculated its distance from the 
earth ; its real diameter and magnitude ; the density of the matter of which it is composed; and the 
laws of the fall of heavy bodies on its surface. 

In the year 1779, there was a spot on the sun, which was large enough to be seen by the naked 
eye ; it was divided into two parts, and must have been 50000 miles in diameter; this phenomenon 
maybe accounted for, from some natural change of an atmosphere. For if some of the fluids which 
enter into its composition be of a shining brilliancy, whilst others are merely transparent, then any 
temporary cause which should remove the lucid fluid, will permit us to see the body of the sun through 
the transparent ones. If an observer were placed on the moon, he could see the solid body of the 
earth only in those places where the transparent fluids of our atmosphere would permit him. In 
others, the opaque vapours would reflect the light of the sun without permitting his view to penetrate 
the surface of our globe. He would probably find that our planet had occasionally some shining fluids 
in its atmosphere, such as the northern lights. And there is good reason to believe, that all the plan¬ 
ets emit light in some degree; for the illumination which remains on the moon in a total eclipse 
cannot be entirely ascribed to the light which may reach it by the refraction of the earth’s atmos¬ 
phere. For, in some cases, as in the eclipse of 1790, the focus of the sun’s rays refracted by the 
earth’s atmosphere, must be many thousand miles beyond the moon. 

There are appearances also which denote a phosphoric quality in the atmosphere of Venus. 

Dr. Herschel supposes, that the spots in the sun are mountains on its surface, which, considering 
the great attraction exerted by the sun upon bodies placed at its surface, and the slow revolution 
it has about its axis, he thinks may be more than 300 miles high, and yet stand very firmly. He says, 
that in August, 1792, he examined the sun, with several powers from 90 to 500. And it evidently 
appeared, that the black spots are the opaque ground or body of the sun; and that the luminous part 
is an atmosphere, which being intercepted or broken, gives us a glimpse of the sun itself. 

Hence he concludes, that the sun has a very extensive atmosphere, which consists of elastic fluids 
that are more or less lucid and transparent; and of which the lucid ones furnish us with light. This 
atmosphere, he thinks, is not less than 1843, nor more than. 2765 miles in height; and he supposes 
that the density of the luminous solar clouds need not be exceedingly more than that of our aurora 
borealis, in order to produce the effects with which we are acquainted. 

The sun, then, appears to be a very eminent, large, and originally luminous body, and the only- 
one belonging to our system. Its similarity to the other globes of the solar system, with regard to 
its solidity;—its atmosphere;—its surface diversified with mountains and vallies ;—its rotation on 
its axis;—and the fall of heavy bodies on its surface,—leads us to suppose that it is most probably 
inhabited, like the rest of the planets, by beings whose organs are adapted to the peculiar circumstan¬ 
ces of that vast globe. 


OF ASTRONOMY. 


Book VII. Part III. 


If it be objected, that from the effects produced at the distance of 97,000,000 miles, we may infer 
that everything must be scorched up at its surface; we reply, that there are many facts in natu¬ 
ral philosophy which show that heat is produced by the sun’s rays only when they act on a calorific 
medium; they are the cause of the production of heat by uniting with the matter of tire which is 
contained in the substances that are heated; as the collision of the dint and steel will inflame a maga¬ 
zine of gunpowder, by putting all the latent tire which it contains into action. 

On the tops of mountains of sufficient height, at the altitude where clouds can seldom reach to shel¬ 
ter them from the direct raj s of the sun, we always find regions of ice and snow. Now if the solar 
rays themselves conveyed all the heat we find on this globe, it ought to be the hottest where their 
course is the least interrupted. Again ; our aeronauts all confirm the coldness of the upper regions oi 
the atmosphere ; and since, therefore, even on cur earth the heat of the situation depends upon the readi¬ 
ness of the medium to yield to the impression of the solar rajs, we have only to admit, that on the sun itself, 
elastic fluids composing its atmosphere, and the matter on its surface, are of such a nature as not to be 
capable of any extensive affection of its own raj^s ; and this seems to be proved by the copious emission 
of them ; for if the elastic fluids of the atmosphere, or of the matter contained on the surface of the sun, 
were of such a nature as to admit of an easy chemical combination with its rays, their emission would 
be very much impeded. Another well known fact is, that the solar focus of the largest fens, thrown 
into the air, will occasion no sensible heat in the place where it has been kept for a considerable time, 
although its pow er of exciting combustion, when proper bodies are exposed, should be sufficient to iuse 
the most refractory substances. 

It is by analogical reasoning that we consider the moon as inhabited. For it is a secondary planet 
of considerable size, its surface is diversified, like that of the earth, with hills and valleys. Its situa¬ 
tion, with respect to the sun, is much like that of the earth; and by a rotation on its axis, it enjoys an 
agreeable variety of seasons, and of day and night. To the moon, our globe would appear a capital 
satellite, undergoing the same changes of illumination as the moon does to the earth. The sun, planets, 
and the starry constellations of the heavens, will rise and set there as they do here; and heavy bodies 
will fall on the moon as they do on the earth. There seems, then, only to be wanting, in order to 
complete the analogj r , that it should be inhabited like the earth. 

It may be objected, that, in the moon, there are no large seas; and its atmosphere (the existence 
of which is doubted by many) is extremely rare, and unfit for the purposes of animal life ;—that its 
climates, its seasons, and the length of its days and nights, totally differ from ours;—that without dense 
clouds, which the moon has not, there can be no rain, perhaps no rivers and lakes. 

In answer to this, it may be observed, that the very difference between the two planets strengthens 
the argument. We find even on our globe, that there is a most striking dissimilarity in the situation of 
the creatures that live upon it. While man walks on the ground, the birds fly in the air, and the fishes 
swim in the water. We cannot surely object to the conveniences afforded by the moon, if those that 
are to inhabit its regions are fitted to their conditions as well as we on this globe of ours. The analogy 
already mentioned establishes a high probability that the moon is inhabited. 

Suppose, then, an inhabitant of the moon, who has not properly considered such analogical reason¬ 
ings as might induce him to surmise that our earth is inhabited, were to give it as his opinion, that the 
use of that great body, which he sees in his neighbourhood, is to carry about his little globe, in order 
that it may be properly exposed to the light of the sun, so as to enjoy an agreeable and useful variety 
of illumination, as well as to give it light by reflection, when direct light cannot be had. Should we 
not condemn his ignorance and want of reflection ? The earth, it is true, performs those offices which 
have been named, for the inhabitants of the moon, but we know that it also affords magnificent dwell¬ 
ing-places to numberless intelligent beings. 

From experience, therefore, we affirm, that the performance of the most salutary offices to inferior 
planets, is not inconsistent with the dignity of superior purposes; and in consequence of such ana¬ 
logical reasonings, assisted by telescopic views, which plainly favour the same opinion, we do not hesi¬ 
tate to admit that the sun is richly stored with inhabitants. 

This way of considering the sun, is of the utmost importance in its consequences. That stars are 
suns can hardly admit of a doubt. Their immense distance would effectually exclude them from our 
view, if their light were not of the solar kind. Besides, the analogy may be traced much farther; the 
sun turns on its axis; so does the star Algol; so do the stars called |3 Lyrae, <1 Cephei, jj Antinoi, a Ceti, 
and many more, most probably all. Now from what other cause can we, with so much probability, ac¬ 
count for their periodical changes ? Again ; our sun’s spots are changeable—so are the spots on the 
star o Ceti. But if stars are suns, and suns are inhabitable, we see at once what an extensive field for 
animation opens to our view. 

It is true, that analogy may induce us to conclude, that since stars appear to be suns, and suns, ac¬ 
cording to the common opinion, are bodies that serve to enlighten, warm, and sustain a system of plan- 


Book VII. Part III. 


OF THE FIXED STARS. 


/ 


231 


ets, we may have an idea of numberless globes that serve for the habitation of living creatures. But 
if these suns themselves are primary planets, we may see some thousands of them with the naked eyes, 
and millions with the help of telescopes; and, at the same time, the same analogical reasoning still 
remains in full force with regard to the planets which these suns may support. 

Moreover, from the observations of Dr. Herschel, on the compressed clusters of stars, it appears, 
that in many instances there cannot be space for the revolutions of a system of planets and comets, and 
therefore it is highly probable that these suns are capital primary planets, which exist for themselves 
and are connected together in one great system for mutual support. See a very curious and valuable 
paper on the nature and construction of the sun and fixed stars, by Dr. Herschel ; read to the lloyal 
Society, Dec. 18, 1794. From this paper, the foregoing scholium has been taken. See also Dr. 
HerschePs paper on the periodical star, « Herculis; with remarks, tending to establish the rotatory 
motion of the stars on their axes. Phil. Trans. 1796. 


1 


/ 


/ 


1 


i 



APPENDIX TO THE ASTRONOMY. 


CONTAINING SOLAR AND LUNAR TABLES, WITH THEIR EXPLANATION AND 
USE, AND THE PROJECTION OF ECLIPSES, SELECTED FROM 
“ EWING’S PRACTICAL ASTRONOMY.” 


EXPLANATION OF THE TABLES. 

Tables of the Mean Motions of Celestial Objects. 

The idea that the sun, moon, and stars, performed all their motions in circles, was, perhaps, one of 
the first which men received concerning these very distant bodies. The regular returns of day and 
night, of the seasons of the year, and of almost all things in the visible world, would serve to confirm 
it; and although it is well known that the orbits of the planets are not perfect circles, it is equally 
known that they differ very little from circles; and therefore modern astronomers retain the idea, and 
form tables of the motions of the planets, as if their orbits were circles, and their motions always uni¬ 
form, passing over equal spaces in equal times. Such are called Tables of Mean Motion ; and the lon¬ 
gitude of a planet computed from such Tables for any given time is called its mean longitude. 

Table I. II. Contain the Sun's .Mean Motion , and the Precession of the Equinoctial Points in Julian 
Years. 

The astronomical year is that space of time wherein the earth moves round the sun, or wherein the 
sun seems to move round the whole ecliptic from any point, such as the vernal equinox, to the same 
again ; which, according to the most accurate observations, consists of 365 natural days, 5 hours, 48 
minutes, 43^ seconds. But in civil reckoning there are two kinds of years, common and bissextile. A 
common year consists of 365 days, and the bissextile of 366. These are called Julian Years , from Julius 
Caesar, who introduced this method of computation. Mr. Mayer makes the sun’s mean motion in 
365 days to be 11s. 29° 45' 40".7. Admitting his numbers, we have, 

For the 4th year, 


Year. 

S. 

o 

/ 

tr 


S. 

o 

1 . 

It 

1 

11 

29 

45 

40.7 

To the 3d year 

11 

29 

17 

2.1 

2 

11 

29 

31 

20.4 

Add mo. for 1 year 

11 

29 

45 

40.7 

3 

11 

29 

17 

2.1 

And for 1 day 



59 

8.3 

4 

0 

0 

1 

51.1 











The 4th year = 

0 

0 

1 

51.1 


That is, in four years the earth goes four times round the sun, and 1' 51'' more. In the same manner, 
the sun’s mean motion may be found for any number of Julian years, as in the Table. 

The yearly motion of the earth’s or sun’s apogee is found to be 1' 6"; which being subtracted 
from the sun’s yearly motion in longitude, the remainder is the sun’s mean anomaly for 1 year; and for 
any number of years the mean anomaly is found in the same manner as the mean longitude. 

It has been found by observation, that the equinoctial points move backward, or contrary to the 
order of the signs, at the rate of 50".3 yearly, called the precession of the equinoxes ; and multiplying 
50".3 by 2, 3, 4, 5, &c. the numbers in Table II. are found. 

Table III. Contains the Sun's Mean Longitude and Anomaly , with the Obliquity of the Ecliptic , for years 
current of the Christian Era. 

The table is composed in this manner: The mean longitude and anomaly of the sun with the ob¬ 
liquity of the ecliptic, are found for the noon of the last day of Dec. 1760, which is accounted his mean 
longitude, &c. for the succeeding year 1761. To these numbers add the sun’s mean motions for 20 

30 




234 


\ 




APPENDIX TO THE ASTRONOMY. 

years, taken from table I; the sums are the sun’s mean longitude and anomaly for 1781. To these add 
the mean motions for 10 years, and the mean longitude and anomaly for 1791 are known. And for 
the following years add the sun’s mean motions in one year to his mean longitude and anomaly for the 
preceding year continually ; remembering to add the motion of 1 day more for every bissextile year 
until the number of years which the table is to contain be completed. 

The annual differences of the obliquity of the ecliptic are very small, and not always regular. It 
appears by the Table that the difference is only about 25" in 60 years ; therefore the obliquity of the 
ecliptic may be stated at 23° 28' during the next fifty years without sensible error. 

When the sun’s mean longitude and anomaly are wanted for any year which is not in the Table, 
take the mean motions for the intermediate years from Table I. and if the year is before 1761, sub¬ 
tract them; if after it, add them to the numbers for 1761, and the remainder or sum will be the mean 
longitude and anomaly required. 

In these additions 12 signs are to be rejected as often as they occur; and in the subtractions 12 
signs must be borrow'ed when necessary. 

Table IV. Contains the sun’s mean motions for the days of the year, distributed into 12 calendar 
months. At the noon of January 1st there is one astronomical day past, because it began at the noon 
of December 31st; and therefore the sun’s mean motions in one day are put down. These being multi¬ 
plied by 2, 3, 4, 5, &,c. (allowance being made for the fractions in one day’s motion) the products are 
the numbers of the Table ; remembering to place them properly, viz. the motions for 31 days at the 
last day of January; for 32 days at the 1st day of February ; and so on until the 31st of December. 

Table V. Contains the sun’s mean motions for hours, minutes, and seconds, which may be under¬ 
stood from what has been said of his motions for days. 

Table VI. Equations of the Sun’s Centre. 

The sun’s mean and true longitudes differ more or less in every point of the earth’s orbit except 
two, viz. the aphelion and perihelion ; and these differences are called equations of the sun’s centre. 
One cause of the difference is, that the earth’s orbit is an ellipse, in the periphery whereof the true 
longitude is reckoned from the vernal equinox, and the mean longitude is reckoned from the same 
point in the circumference of a circle, whose diameter is the greater axis of the ellipse. The circle 
and ellipse coincide only in the aphelion and perihelion ; and therefore when the sun or earth is in eith¬ 
er of these points, the mean and true longitudes are the same, and there is no equation. 

Another cause of the difference between the mean and true longitudes arises from the unequal mo¬ 
tion of the earth in its orbit; for while the earth proceeds from its aphelion to its perihelion, its motion 
is continually accelerated; and from its perihelion to its aphelion, its motion is continually retarded ; 
and this is true of every planet. 

When the sun appears in the earth’s aphelion, his longitude is about 3s. 9°, and his anomaly is 0, 
because it is reckoned from that point; and when he appears in the earth’s perihelion, his longitude 
is nearly 9s. 9°, and his anomaly is 6 signs. In the first six signs of anomaly, the equation found in 
the Table is to be subtracted from the mean longitude ; in the other six signs it is to be added, and 
the remainder or sum is the true longitude. 

Table VII. Contains the Logarithms of the Earth's Distances from the Sun. 

Because the earth’s orbit is an ellipse, and the sun in one of its foci, the earth’s distance from the 
sun varies every moment; for it is greatest in aphelion, and decreases as the earth proceeds from 
thence toward the perihelion, where it is least. The several distances decrease in the first six 
signs of anomaly, and increase in the other six. When the anomaly is 3 or 9 signs, the earth is at its 
mean distance from the sun expressed by 1000000. In all other points the distance is either greater or 
less than the mean. m 

The earth’s distance from the sun being calculated for every degree of anomaly, the logarithms of 
these distances are contained in the Table, and are of use in computing the longitudes of the other planets. 

Table VIII. The Sun's Declination to every Degree of his Longitude. 

At the time of either equinox the sun is in the equinoctial circle, but at all other times he appears 
at some distance from it, either north or south ; and this distance is called his declination. 

At the vernal equinox the sun has no declination; but from that point his declination increases 
northward, until he comes to the summer solstice, where it is greatest ; and from thence it decreases 
until the sun is at the autumnal equinox, when again it is nothing; and then changes its name from north 
to south, and increases southward to the winter solstice, when again it is greatest; and from thence it 
decreases until the sun appears in the vernal equinox. 




) 


EXPLANATION OF THE TABLES. 


235 


Those points of the ecliptic, which are equally distant from the equinoxes or solstices, being also 
equally distant from the equinoctial circle, have the same declination ; and therefore the declination being 
calculated lor every degree of the first quadrant of the ecliptic, answers for the whole; for the begin¬ 
ning of the sign Taurus, of Virgo, of Scorpio, and of Pisces, are ail at the sarr^i distance from the equi¬ 
noctial circle, and consequently these points have the same declination ; only in the two first the declina¬ 
tion is north, and in the two last it is south; and the same is true of all other points of the ecliptic which 
are equally distant from the equinoctial or solstitial points. 

Table IX. The Sun’s Apparent Semidiameter and Hourly Motion. 

\ 

When the earth is in its aphelion, the sun’s diameter appears least, and his apparent motion slowest; 
his diameter at that time is only 31' 34'', and his hourly motion 2' 23". While the sun moves appar¬ 
ently from the aphelion to the perihelion, his diameter and hourly motion increase with his anomaly, 
and are greatest in the perihelion ; his diameter being then 32' 38", and his hourly motion 2'33". The 
difference between the aphelion and perihelion diameter is 64", and between the hourly motions 10''. 

The Table contains the sun’s semidiameter and hourly motion to every 10° of mean anomaly ; which 
by taking proportional parts for the intermediate degrees, will serve to find them for any given anomaly. 
They are frequently of use in Astronomy, especially in calculating eclipses. 

Table X. Contains the equation of time for every degree of the sun’s longitude ; and by using pro¬ 
portion for the minutes and seconds, the equation may be found for any given longitude. For example, 
let it be required to find the equation of time when the sun’s longitude is 6s. 10°. 50' 30". Ans. 11m. 
12s. 


o 


m. 

s. 

0 s. ' " 

s. 

10 

gives 

10 

57 

As 1 : 18 :: 50 30: 

15 

11 

11 

15 

Add - 10 

57 

Diff. 


- 

18 

True equation 11 

12 

Table 

XI. 

The Sun's Longitude every day at JYoon. 



The conveniency of having the sun’s longitude nearly true for the noon of every day in the year 
inclined me to insert this Table. The reason why it could not be made accurate is, a common year 
of 365 days differs from the time wherein the sun seems to move round the whole ecliptic by almost 6 
hours, or a day in four years ; which causes a difference in the sun’s longitude at noon on the same days 
of different years ; and therefore, to have made the table more perfect, it must have been calculated for 
four years, which was not thought expedient. 

Tables of the Moon’s Mean Alotions. 

These Tables are made in the same manner as those of the sun or earth, the moon's period being 
known; but her motions are more in number than those of the earth; for besides her mean longitude 
and anomaly, the longitude of her node must be known, in order to calculate her place for any given 
time. 

The earth is in one of the foci of the moon’s orbit, and is the centre of her motion ; for the moon re¬ 
volves round the earth in the same manner as the earth moves round the sun. 

The moon’s period, or the time wherein she moves once round the earth, is 27 days, 7 hours, 43 min¬ 
utes, 5 seconds; and to find the moon’s mean motion in a common year of 365 days the proportion is, 

As the moon’s period 27d. 7h. 43m. 5s. 

Is to her whole orbit or 360°; 

So is a common year of 365 days 
To 13 revolutions, and 4s. 9°. 23' 5". 

The 13 revolutions are rejected, and 4s. 9°. 23' 5" are taken for the moon’s motion in 365 days. 

For the Mooids JWean Motion in one day. 

As the moon’s period 27d. 7h. 43m. 5s. 

Is to the whole circle or 360°; 

So is one day 

To the'moon’s mean motion in 1 day 13° 10' 35". 

Having the moon’s mean motion in 1 year and 1 day, the Table of her mean longitudes for Julian 
years is made by multiplying the motion for 1 year by 2, 3, 4, &c. and to the 4th adding the motion for 1 
day, the products are the mean longitudes for these years ; and by multiplying the motion for 1 year by 




236 


APPENDIX TO THE ASTRONOMY. 


the several numbers of years, and adding the motion in 1 day to every 4th or bissextile year, the Table 
may be made to any extent. 

To calculate the Moon’s Mean Anomaly. 

The moon’s apogee moves once round her whole orbit in 8 years 309 days 8 hours 20 minutes ; or 
(adding 2 days for leap years) in 3231 days 8 hours 20 minutes. Then, 

As 3231d. 8h. 20m. 

Is to the whole circle of 360°; 

So is a common year ot 365 days 

To the mo. of the J) ’s apogee in 1 year 40° 39' 50''. 

s. ° ' " 

From the ^)’s m. mot. in Ion. during 1 year 4 9 23 5 
Subtract the motion of her apogee in ditto —1 10 39 50 


There remains the D’s mean anomaly in 1 year 2 28 43 15 
To find the Moon’s Mean Anomaly for one Day. 

Divide the motion of her apogee in a year, viz. 40° 39' 50'' by 365, and the quotient 6' 41" is the 
motion of the moon’s apogee for 1 day. 

o r n 

From the moon’s motion in longitude for 1 day 13 10 35 

Subtract the motion of her apogee for ditto — 6 41 

There remains the moon’s mean anom. for 1 day 13 3 54 

Having found the moon’s mean anomaly for 1 year and 1 day, the several mean anomalies in the Table 
are found in the same manner as the longitudes, viz. by multiplying the mean anomaly for 1 year by 2, 
3, 4, &.c. and adding the motion for 1 day to every 4th or bissextile year. 

To find the Mean Motion of the Moon’s Mode. 

The moon’s node moves backward round her whole orbit in 18 years 224 days 5 hours; therefore, 
for its motion in one year, 

As 18 years 224 days 5 hours 
Is to the whole circle or 360°, 

So is a year of 365 days 

To the motion of the moon’s node in 1 year 19° 19' 43''. 

For the motion of the moon’s node in 1 day, divide 19° 19' 43" by 365, and the quotient, which is 
nearly 3' 11'', is its motion for one day. 

The motion of the moon’s node for 1 year and 1 day being known, its motion for any number of 
years is found in the same manner as those of mean longitude and anomaly, viz. by multiplying one 
year’s motion by 2, 3, 4, &.c. and adding the motion of a day to every fourth or bissextile year. 

Table II. Containing the Moon’s Mean Longitude and Anomaly , with the Longitude of her Node, for current 
Years. 

To make the Table, find, either by calculation or a good observation, the moon’s mean longitude 
and anomaly, with the longitude of her node, on the noon of the last day of December preceding the 
year, where the Table is to begin, which here is the year 1760, and these are the mean places for 
3761, the first in our Table. For 1781, the next year in the Table, take the mean motions for 20 vears 
from Table I. and add the longitude and anomaly to those for 1761, but subtract the motion of the node; 
the results are the mean places for 1781 ; then take the mean motions for 10 years, and apply them in 
the same manner to the numbers for 1781, and the mean places for 1791 will be known. 

For the following years the Table is carried ou by the continual addition of the mean longitude and 
anomaly for one year, as also the motion for 1 day more every 4th or bissextile year, and subtracting 
the motion of the node. 


. Table III. Contains the Moon’s Mean Motions for Days. 

It begins with the mean motions for one day already found. These being multiplied by every num¬ 
ber of days from 1 to 365, place the products at the proper days of the several months, and the table 
is made. 




EXPLANATION OF THE TABLES. 


237 


Table IV. Contains the Moon’s Mean Motions for Hours , Minutes , and Seconds. 

The numbers are found by dividing the mean motions for 1 day by 24, the quotients are the mean 
motions for an hour; and these again divided by 60, give the mean motions for 1 minute, &c. The 
motions for the different numbers of hours and minutes are found by multiplication. 

The moon’s motions are affected with many inequalities, and therefore many equations are neces¬ 
sary to reduce her mean longitude to the true. The method of finding the arguments, of taking out 
the equations, and applying them, is described in Prob. VII. 

The Tables for finding the moon’s latitude, parallax, diameter, hourly motion in longitude and lat¬ 
itude, at any given time, are sufficiently plain from their titles; and the method of applying them is 
described in Prob. VIII—XIII. 

Tables of the Mean Motion of the Moon from the Sun. 

The moon’s mean motion in a common year of 365 days is 4s. 9° 23' 5'', over and above 13 revolu¬ 
tions ; and the sun’s apparent mean motion in the same time is 11s 29° 45' 40". 

S. ° ' " 

From 4 9 23 5 

Subtract 11 29 45 40 


The remainder 4 9 37 25 is the moon’s mean motion from the sun in a common year. 

In the same manner, the moon’s mean motion from the sun may be found for a day, an hour, or for 
any number or parts of these. 

Having the moon’s mean motion from the sun in a year and a day, it may be found for any number 
of years by multiplication ; remembering to add the motion for one day to every four years’ motion for 
leap year. 

To make a Table of the moon’s mean motion from the sun for current years of the Christian era, 
subtract the sun’s mean longitude from the moon’s for every year which the Table is to contain ; and 
the remainders will be the numbers expressing the moon’s mean motion from the sun ; as in the Table. 

The table for months is made by subtracting the sun’s mean longitude from the moon’s on the last 
day of the preceding month ; and the remainder is the moon’s mean motion from the sun on the first 
day of the following month. The Tables for days, hours, minutes, and seconds, are made in the same 
manner; and the method of using these Tables is described in Prob. XIV. 

Table of Mean New Moons in March , with the Mean Anomalies of the Sun and Moon ; and the Sun’s Mean 
Distance from the Moon’s Node. 

To make Table I. Calculate the time of mean new moon in March, for the year with which the 
Table is to begin, by the rule, Prob. XIV. or by any other method. Calculate also the sun and moon’s 
mean anomalies for that time, and the sun’s mean distance from the moon’s node, from the Tables of 
mean motion, and write them down in order. 

For the following years. If the new moon falls after the 11th day of March, add 12 lunations to the 
time for the former year; and if the next is a common year, subtract 365 days from the sum; the re¬ 
mainder is the time of mean new moon in March the next year; but if the next be a leap year, subtract 
366 days. If the mean new moon falls before the 11th of March, add 13 mean lunations, and subtract 
365 or 366 days from the sum, according as the next is a common or bissextile year. 

Calculate also the sun and moon’s mean anomalies, with the sun’s mean distance from the moon’s 
node for 12 or 13 lunations, and add them to those of the former year; and the numbers for the second 
year in the Table will be known. Proceed in the same manner foV every succeeding year until the 
Table is completed. 

Table II. Contains \2>\ Mean Lunations , with the Anomalies , and the sun’s Mean Distance from the Moon’s 
Node. 

The numbers are computed from the Tables of mean motion thus : Take the mean anomalies of the 
sun and moon out of the Tables for 29 days 12 hours 44 m. 3 s. ; take also the mean motion of the sun, 
and of the moon’s node, out of their proper Tables for the same time ; their sum is the sun’s mean 
distance from the moon’s node. Having found the numbers for one lunation, multiply them by 2, 3, 4, 
&c. until there are 13 products, and divide the numbers for one lunation by 2 for the half lunation. 

By these Tables the times of mean syzygies for any month of a given year, within the limits of 
Table I. may be found. 

The Tables of equations for reducing the mean to the true times of new and full moons, in the fol¬ 
lowing pages, depend on the mean anomalies of the sun and moon, with the sun’s mean distance from 



238 


APPENDIX TO THE ASTRONOMY. 


the moon’s node, and are expressed in time for the conveniency of calculation. The method of apply’ 
ing them is given in Problem XVI. 

Construction and Use of Logistical Logarithms. 

Logistical Logarithms are artificial numbers deduced from common logarithms, and made for min¬ 
utes and seconds either of degrees or time, used in working proportions, wherein some or all of the 
given terms are sexagesimals. 

The numbers which compose the Table of logistical logarithms are regulated by its extent. For if 
the Table contains no more than the minutes and seconds in 1°, or 3600'', the logistical logarithm of 
1" is the logarithm of 3600', viz. 3.5563; and all the following numbers are the differences between 
3.5563 and the logarithms of the several numbers of seconds from 1 to 3600". Therefore to calculate 
the logistical logarithm of any quantity less than 1°, subtract the logarithm of the given number of 
seconds from 3.5563 ; the remainder is the logistical logarithm required. 

Exam. Let it be required to find the logistical logarithm of 10', or 600". 

From the common log. of 3600 = 3.5563 
Subtract the common log. of 600 = 2.7781 

i - 

There remains the logistical log. of 10' = 0.7782 

But if the Table is to contain more than 1° such as in Dr. Maskelyne’s Proportional Logarithms, 
which extends to 3°, or 10800''; then, in such a Table, the proportional or logistical logarithm of V is 
the logarithm of 10800, viz. 4.0334 ; and all the rest are the differences between 4.0334 and the log¬ 
arithms of the several numbers of seconds from 1 to 10800". 

Exam. Let it be required to find the logistical or proportional log. of 10' or 600". 

From the log. of - 10800 =4.0334 

Subtract the log. of - 600 = 2.7781 


Remains the prop. log. of 10' 1.2553 

Hence there may be as many different systems of logistical or proportional logarithms as any one 
chooses to assume different extents of the Table. Our Table extends only from 1" to 1°, which is suffi¬ 
cient for common use. Logistical logarithms consist of four figures beside the index. 

The numbers on the head of the Table are either degrees or minutes; and those in the left hand 
column, on the side, are minutes or seconds; but if the numbers on the head be hours or minutes, those 
in the left hand column, on the side, will be minutes or seconds of time. Therefore these numbers 
change their denomination as occasion requires. The second line on the head of the Table are the 
numbers of seconds in the minutes which stand over them. 

To take the logistical logarithms of any number of minutes and seconds out of the Table, the given 
number must be within the limits of the Table. 

Find the minutes on the head of the Table, and the seconds in the left hand column on the side, 
and under the minutes and opposite to the seconds stands the logistical logarithm required. 

Examples. 


Given numbers 0 

40 

Logist. log. 1.9542 

1 

10 

1.7112 

7 

40 

8935 

# 9 

15 

8120 

50 

0 

792 

50 

37 

739 


A logistical logarithm being given, to find the number of minutes and seconds answering. 

If the given logarithm be found in the Table, the minutes are on the head of the column and the 
seconds on the side ; but if it be not found exactly in the Table, take the next greater than it, and the 
minutes and seconds answering are found on the head and side of the table, as before. 

Examples 


L. Log. 

Values. 

/ 

1.7112 

1 10 

9823 

6 15 

6746 

12 41 

5175 

18 13 




EXPLANATION OF THE TABLES. 


239 


When the logistical logarithm of any number less than 3600 is required, find it, or the nearest less, 
in the second line on the head of the Table ; and below the number on the head, and opposite to its 
excess on the side, stands the logistical logarithm. Thus the logistical logarithm of 1276 is 4505. 

Logistical logarithms are used in finding a fourth proportional, when some or all of the given terms, 
are minutes and seconds either of motion or time; and the method of operation is the same as in work¬ 
ing with common logarithms, that is, by adding the logarithms of the second and third terms, and sub¬ 
tracting the logarithm of the first term from the sum, the remainder is the logarithm of the fourth term, 
or answer. 

Exam. 1 . Suppose the sun’s hourly motion was 2' 27", and it were required to find his motion in 
47' 10". The answer is T 55". 

As 60 m. 0 

To 2' 27" 13890 

So 47 m. 10s. 1045 


To 1' 55" 14935 

2. Suppose the hourly motion of the moon from the sun is 32' 8", and the moon is 58' 12" behind 
the sun. In what time will the moon overtake the sun ? Jins. 1 h. 48 m. 40 s. 

JV. B. In such questions, when the fourth proportional would exceed 60 m. which is beyond the 

limit of the Table, take or I, of the third term, and multiply the fourth term, when found, by 2, 

3, or 4. 

As 32' 8" 2712 

To 60 m. 0 

So 29' 6" 3143 


To 54 m. 20 s. 431 

2 


Doubled is 1 h. 48 m. 40 s the answer. * 

In like manner, when two of the given terms exceed the limits of the Table, divide all the terms 
by some number, such as 2, 3, or 4, and multiply the answer by the. same number which divided the 
given terms. Half of each term. 


Suppose we have, As 76' 34" 38' 17" 1951 

Is to 24 h. 12 h. 6990 

So is 64' 20" 32' 10" 2707 

To 20 h. 10 li nearly 10 h. 5 m. 9697 

Doubled 20 h. 10. m. 7746 


PROBLEMS, 

Showing the Use and Application o f the Tables and Projection of Eclipses . 


PROB. I. The Longitudes of two Places , and the Time at one of them being given , to find the corresponding 
Time at the other. 

Reduce the difference of the longitudes to time, by allowing at the rate of 15° to an hour, and add 
it to the given time for a place toward the east; but subtract it for a place toward the west. 

Exam. 1 . Suppose the time at Greenwich to be 6 h. 7' 8 " P. M. required the corresponding time 
at Cambridge, in longitude 71° 7' 25" west. f 

Time at Greenwich 6 7 8 P. M. 

Diff. long. 71° T 25" = — 4 44 29f 

Time at Cambridge 1 22 38^ P. M. 

Exam. 2. The time at Cambridge being 7 h. 43' 57" A. M. required the corresponding time at 
Greenwich. 


1 


i 











240 



APPENDIX TO THE ASTRONOMY. 

h. ' " 

Time at Cambridge 7 43 57 A. M. 

Diff. long. 71° 7' 25" = -f 4 44 29§ 


Time at Greenwich 28 26| P. M. 

Exam. 3. The time at Hamburgh, in longitude 10° 38' E. being 6 h. 30' 40 A. M. required the cor¬ 
responding time at Cambridge, in longitude 71° 7' 25" W. 

h. ' " 

Time at Hamburgh 6 30 40 A. M. 

Diff. long. 81° 45' 25" 5 27f 

Time at Cambridge 1 3 38^ A. M. 

PROB. II. To calculate the true Longitude of the Sun for any given Time and Place by the Tables. 

When the given time is apparent, reduce it to mean time, and if the given place be not in the me¬ 
ridian of the Tables, that is, Greenwich, reduce it to that meridian by Prob. I. and then write down 
the years, months, days, hours, &c. under one another in a column on the left hand. 

Take out of the Tables the sun’s mean longitude and anomaly, answering to each part of the time, 
and write them down on the right hand of the former numbers'; then add them as they stand in their 
several columns, rejecting 12 signs, or any multiple thereof, and the sum will be the mean longitude of 
the sun for the given time. 

To reduce the mean longitude to the true. Enter the Table with the sun’s mean anomaly, and 
take out the equation of the centre, making proportion for the odd minutes, &c. and if the sign of 
the sun’s mean anomaly be at the head of the Table, subtract the equation from the mean longitude , 
but if it be at the foot of the Table, add it to the mean longitude, and the true longitude of the sun 
will be had. 

Exam. 1 . Required the sun’s longitude 1796, Feb. 19, 11 h. 21' 30" in the forenoon, apparent time, 
in the meridian of Edinburgh. 




D. 

h. 

/ 

V 

Given time February 

- 

18 

23 

21 

30 

Edinburgh meridian W. 

- 

+ 


12 

25 

Equation of time 

- 

+ 


14 

14 

Mean time at Greenwich 

18 

23 

48 

9 


M. Long. 



M. Anom. 


S. 0 ' 

n 


S. 0 

/ n 

1796 

no 5i 

52 


6 1 

23 42 

Feb. 18. bissex. 

1 17 18 40 


1 17 

18 31 

23 h. 

56 

40 



56 40 

48' 

1 

58 



1 58 

9" 

10 29 9 

10 


7 19 

40 51 

Equation of the centre 

-j- 1 -28 

30 




Sun’s true longitude 

11 0 37 

40 





2. Required the sun’s longitude 1795, June 22, at noon, apparent time, at Greenwich. 

h. ' " 

Mean time - - - 0 0 0 

Equation of time - + 1 28 




0 

1 28 


M. Long. 

M. Anom. 


S ° ' 

n 

S. ° ' " 

1795 

9 10 7 

2 

6 0 39 58 

June 22, 

5 20 31 

1 

5 20 30 30 

Oh. 1' 

- 

2 

2 

28" 

- 

1 

1 


3 0 38 

6 

11 21 10 31 

Equation 

of the centre -f- 17 

23 



Sun’s true longitude 3 0 55 29 












USE AND APPLICATION OF THE TABLES, &c. 

PROB. III. The Mean Anomaly of the Sun being given , to find the Logarithm of his Distance from the Earth. 

With the mean anomaly of the sun take out the logarithm of his distance from the earth from the 
Table. 

Exam. Let the sun’s mean anomaly be 7s. 19° 40' 51'', what is the logarithm of his distance from the 


earth ? 


For 7s. 19° the log. is 


4.995224 



For 40' 5l'' the part is 

- 

-f 69 



Answer 


4.995293 


PROB. IV. To find the Apparent Semidiameter , and also the Hourly Motion of the Sun for any given Time. 

Calculate the sun’s mean anomaly for the given time, and with the mean anomaly of the sun take 
out his semidiameter and hourly motion from the Table. 

Exam. Required the sun’s apparent semidiameter and hourly motion 1796, February 19th, llh. 21' 
in the forenoon. 

The sun’s mean anomaly at that time is 7s. 19° 42', which gives his semidiameter 16' 13" and his 
hourly motion 2' 31". 

PROB. V. The Longitude of the Sun , or any Point of the Ecliptic , being given , to find the corresponding 

Declination. 

With the given longitude take the declination out of the Table. 

Exam. Let the sun’s longitude be 11s. 0° 37' 40", what is his declination? Ans. 11° 15' 47'' south. 
To longitude 11s. 0° the declination is 11° 29' 5" S. decreasing; the difference between this and 
the next is 21' 12". Then, 



O 

t 

u 

L.L. 




As 

1 

0 

0 

0 


O 

/ n 

To 

0 

21 

12 

4518 

From 

11 

29 5 

So is 

0 

37 

40 

2022 

Subtract 

— 

13 18 

To 

0 

13 

18 

6540 

Declinat. 

11 

15 47 S. 


PROB. VI. The time of the Year and the Sun's Decimation being given , to find his Longitude ; or the 
Declination of a Star , or of any Point of the Ecliptic , being given, to find its Longitude. 

With the given declination take the longitude out of the same Table, as in the preceding problem. 
Thus, take the declination in the Table nearest less than that given, and write down the corresponding 
longitude; marking whether the declination be increasing or decreasing; then subtract the declina¬ 
tion found in the Table from the next greater, the remainder is the difference of declination for 1° of 
longifOde. Subtract also the declination found in the Table from that given ; and then, as the differ¬ 
ence of declination for 1° of longitude is to 1°, so is the difference between the given declination and 
that found in the Table to a part, to be applied to the longitude already found ; the result is the true 
longitude. 

Exam. February 19, 1796, the sun’s declination is 11° 15' 47" S. what is his longitude ? Ans. 11s. 
0° 37' 39 '. 

1st. To declin. 11° 7' 53" S. the long, is 11s. 1°, the declination decreasing. 

o 9 n 

2d. From the given declin. - - - 11 15 47 

Subtract the next less in the Table - 11 7 53 


Difference - 07 54 

3d. Also from the next greater - - 11 29 5 

Subtract the next less in the Table - 11 7 53 


Diff. declin. for 1° long. 

4th. As 
To 
So is 

To 

31 


0 21 12 


O 

/ 

// 

L. L. 

0 

21 

12 

4518 

1 

0 

0 

0 

0 

7 

54 

8805 

- 0 

22 

21 

4287 


241 







V 


12 APPENDIX TO THE ASTRONOMY. 

/ 

To be subtracted. 

g o ' u 

From the longitude found - - 11 1 0 0 

Subtract the prop, part - - —22 21 


There remains the true longitude - - 11 0 37 39 

PRQB. VII. To calculate the true Longitude of the Moon for any given Time by the Tables* 

Calculate the true longitude of the sun and his mean anomaly for the given time by Prob. If. 

From the Tables of the moon’s mean motion, take out the mean longitude, anomaly, and ascending 
node, for the given year; under which write down the mean motions for the month, day, hours, &c. 

Add the numbers in the columns of the mean longitude and anomaly, rejecting 12 signs or any mul¬ 
tiples thereof when they occur; but from the longitude of the node for the given year subtract the sum 
of all the numbers below it, borrowing 12 signs when necessary; and thus the moon’s mean longitude, 
mean anomaly, and the mean longitude of her node, will be obtained. 

For the Arguments of the Equations. 

The sun's mean anomaly is the first argument. Subtract the true longitude of the sun from the mean 
longitude of the moon, the remainder is the mean distance of the moon from the sun, of which take the 
double. 

To and from the double distance of the moon from the sun add and subtract the first argument, the 
sum and remainder are the second and third arguments. 

Add and subtract the moons mean anomaly to and from the double distance of the moon from the 
sun for the fourth and fifth arguments. 

To and from the fifth argument add and subtract the first argument, and the sixth and seventh argu¬ 
ments are found. 

Subtract the first argument from the moon’s mean anomaly, the remainder is the eighth argument. 

Subtract the moon’s mean anomaly from the moon’s mean distance from the sun, the remainder is the 
ninth argument. 

Subtract the true longitude of the sun from the mean longitude of the moon’s node, and the remainder 
is the tenth argument. 

With these arguments take the several equations out of the Tables, and find their sum. 

With the mean anomaly of the sun take out of the Tables the equations of the node and of the moon’s 
mean anomaly. Apply the equation to the longitude of the node, and its equated longitude will be 
known. Correct the mean anomaly of the moon by its own equation, and also by the sum of the ten 
former equations, and the correct anomaly of the moon will be had ; which is the eleventh argument, with 
which take out of the Table the equation of the centre. Correct the mean longitude of the moon both 
by the sum of the ten former equations and by the equation of the centre, and the equated longitude of 
the moon will be known. 

Subtract the sun’s true longitude from the equated longitude of the moon, the remainder is the twelfth 
argument, with which take the variation out of the Table, and apply it to the equated longitude of the 
moon. 

Subtract the correct longitude of the node from the longitude of the moon twelve times corrected, and 
from double of the remainder subtract the moon’s correct anomaly, the remainder is the 13th argument; 
with which take the thirteenth equation from the Table and apply it to the moon’s longitude last found 
which will give the moon’s longitude in her orbit. 

From the moon’s longitude in her orbit subtract the correct longitude of the node, the remainder is the 
fourteenth argument; with which take out of the Table the reduction of the moon’s place in her orbit to 
the ecliptic. And, lastly, with the mean longitude of the moon’s node take the equation of the equinoxes from 
the Table ; and apply both the reduction and the equation of the equinoxes to the moon’s longitude in 
her orbit, according to their signs, and the true ecliptic longitude of the moon will be found. 

Exam. It is required to find the moon’s longitude 1795, June 22, at noon, apparent time, in the merid¬ 
ian of Greenwich. 

The equation of time is -f 1' 28", therefore the mean time is Oh. 1' 28'. 

Sun’s longitude 3s. 0° 55' 29". Mean anomaly, 11s. 21° 10' 31". 



USE AND APPLICATION OF THE TABLES, ,&c. 


D’s m. Long. 

m. Anom. 

.. £ 0 / " 

1795 - 1 5 37 57 

June 22 - 3 29 31 0 

Oh. 1m. 33 

28s. - lb 

5 ° ' " 

6 13 36 29 
3 10 14 35 

33 

15 

5 5 9 45 

Sum of 10 eqs. +19 0 

9 23 51 52 

— 3 18 

5 5 28 45 

Equat. centre + 5 35 20 

9 23 48 34 
+ 19 0 

D’s equat. long. 5 11 4 5 

Variation + 20 48 

9 24 7 34 

• 

5 11 24 53 

13th - + 46 

1) ’s orbit long. 5 11 25 39 
Reduction 6 43 

511 18 56 
Equinoxes — 15 

D’s eclip. long. 5 11 18 41 


a 


Equations. 

S ° ' " 


+ 


4 9 57 43 


/ // 



1 


1 42 

— 9 9 40 

2 


0 48 


3 


0 52 

4 0 48 3 

4 

0 50 


—1 24 

5 

20 36 



6 


0 14 

4 0 46 39 

7 


0 19 


8 


0 36 


9 

1 14 



10 

0 51 






—1 I LIJ. 






+ 23 31 

— 4 31 



— 4 31 




+ 19 0 




Arguments found. 

Arguments found. 


S 

O 

f 

// 


S 

O 

/ 

// 

6th = 

6 

5 

47 

11 


o 

- O 

0 

55 

29 

7th = 

6 

23 

26 

9 

12th = 

2 

10 

8 

36 


9 

23 

51 

52 

D’s long, equat. 

5 

11 

4 

5 


—11 

21 

10 

31 

a 

—4 

0 

46 

39 

8th = 

10 

2 

41 

21 


1 

10 

17 

26 


2 

4 

14 

16 







—9 

23 

51 

52 

Doubled 

2 

20 

34 

52 






D’s Anom. 

—9 

24 

7 

34 

9th = 

4 

10 

22 

24 











13th = 

4 

26 

27 

18 

a 

4 

0 

48 

3 






o 

- —3 

0 

55 

29 

D in orbit 

5 

11 

25 

39 






a - 

—4 

0 

46 

39 

10th = 

0 

29 

52 

34 











14th = 

1 

10 39 

0 

5’s eq. long. 

5 

11 

4 

5 







PROB. VIII. The Moon’s Longitude , Anomaly , and Mode , being given, to find her Latitude by the Tables. 

Subtract the corrected longitude of the moon’s node from the moon’s longitude in her orbit, the re¬ 
mainder is the first argument of latitude. 

From the moon’s longitude in her orbit subtract the sun’s longitude; and from double of the remain¬ 
der subtract the first argument of latitude, the remainder is the second argument. 

Subtract the moon’s mean anomaly from the first argument of latitude, and the remainder is the third 
argument. 

From the third argument subtract the moon’s mean anomaly for the fourth argument; also subtract 
the moon’s mean anomaly from the fourth argument, and the remainder is the fifth argument. 

















































APPENDIX TO THE ASTRONOMY. 


244 


With these arguments take out the numbers with their proper signs from the Tables, and take the 
sum of those which are affirmative, and also the sum of the negative; the difference of these sums, with 
the sign of the greater, will be the latitude of the moon; and will be north if marked with the sign -f-, 
but south if marked with the sign —. 

Exam. It is required to find the moon’s latitude when her longitude in her orbit is 5s. 11° 25' 39", 
mean anomaly 9s. 23° 51' 52", and node 4s. 0° 46' 39". The sun’s longitude is 3s. 0° 55' 29". 


Arguments found. 

Vruments found. 

Equations. 

S ° ' " 

o ° ' 


*+■ 

— 

5’s orbit Ion. 5 11 25 39 

2d =• 3 10 21 20 


Q 1 II 

II 

SI -4 0 46 39 


i 

3 21 23 


- - 

1st - - 1 10 39 0 

2 

0 8 40 


Arg. 1st = 1 10 39 0 

j’s Anom. —9 23 51 52 

o 

O 

}. 


16 

3 

j> - - - 5 11 25 39 

3d = - - 3 16 47 8 

5 


14 

© - . - —3 0 55 29 

—9 23 51 52 



— 


— 


3 30 3 

33 

2 10 30 10 

4th = - - 5 22 55 16 


— 33 



—9 23 51 52 




Doubled - - 4 21 0 20 



+3 29 30 


1st —1 10 39 0 

5th - - - 7 29 3 24 


Ans. The moon’s lat. is3 o 29'30''JX. 


PROB. IX. The Moon's Longitude -with the Arguments for finding it , being known , to find the Moon's 
Equatorial Parallax or the Horizontal Parallax of the Moon for any Place on the Equator. 

With the 5th, 11th, and 12th arguments of longitude, take as many numbers from the Tables, with 
their proper signs ; and from the sum of the affirmative numbers subtract the sum of the negative, the 
remainder is the moon’s equatorial parallax, or the horizontal parallax of the moon to a place on the 
equator. 

Exam. Let the moon’s longitude be 5s. 11° 18' 41", and the arguments as below; required her equa¬ 
torial parallax. Ans. 56' 5". 



+ 

—■ 

S 0 ' " 

/ II 

rl 

5th Arg. 6 14 36 40 

0 36 


11th 9 24 7 34 

55 49 


12th 2 10 8 36 


20 


+ 56 25 



— 20 


Moon’s equatorial parallax 

56 5 



PROB. X. The Equatorial Parallax of the Moon being given and the Latitude of a place , to find the Hori¬ 
zontal Parallax of the Moon at that place ; arid to reduce the Apparent Latitude of the Place to the 
Centre of the Earth , supposing the Eurth is an Oblate Spheroid. 

Enter the Table, with the given apparent latitude of the place on the side, and the equatorial paral¬ 
lax at the top, and making proportion, if necessary, find the reduction of parallax ; which subtracted 
from the equatorial parallax, leaves the horizontal parallax required. 

With the apparent latitude of the place take the reduction of latitude out of the same Table; which 
being subtracted from the apparent latitude, leaves the latitude reduced. 

N. B. The horizontal parallax and latitude thus found are what should be employed in computin°- 
the moon’s parallaxes in longitude, latitude, and declination; which will now be performed by the 
common rules, founded on the supposition of the earth’s being a sphere. 

Exam. Let the equatorial parallax be 56' 5", and the apparent latitude of the place 51° 28' 40"; the 
moon’s horizontal parallax and the reduced latitude of the place are required. 

Off/ 

Latitude 51 28 40 

Reduction — 14 42 


# II 

Equatorial parallax 56 5 
Reduction — 9 


O 


\ 


Horizontal parallax 55 56 


Reduced lat. 


51 13 58 
























245 


USE AND APPLICATION OF THE TABLES, &c. 


PROB. XI. The Equatorial Parallax being given, to find the Moon’s Horizontal Diameter. 

Enter the Table with the moon’s equatorial parallax, and take out her horizontal diameter. 

Exam. The moon's equatorial parallax being 5G' 5", the diameter is 30' 34". 

PROB. XII. To find the Moon's Hourly Motion in Longitude; her Longitude in her Orbit, with the Argu¬ 
ments for finding it, being known. 

With the 5th, 11th, and 12th arguments of longitude, take out the numbers from the Tables; the 
sum of these, regard being had to their signs, is the moon’s hourly motion in longitude. 

Exam. Let the moon’s longitude be 5s. 11° 25' 39", and the arguments as below ; required her hourly 
motion in longitude. 


5 th Arg. 


11th 

12th 



+ 

— 

S 0 ' " 

• /' 

/I 

6 14 36 40 

0 4 


9 24 7 34 

31 20 


2 10 8 36 


31 


+ 32 1 



— 31 


Answer. 

31 30 



PROB. XIII. To find the Moon’s Hourly Motion in Latitude, the Arguments for finding her Latitude being 

known. 

With the first and second arguments of latitude, take two numbers out of the Tables; their sum, if 
they have the same sign, or their difference, if they have contrary signs, is the moon’s hourly motion 
in latitude ; which tends to the north if it has the sign +, but to the south if it has the sign —. 

Exam. It is required to find the moon’s hourly motion in latitude, the arguments of latitude being, 

go in in 


1st Arg. 
2d 


1 10 39 0 

3 10 21 21 


Answer 


+ 2 


15 

1 


+2 14 tending north. 


PROB. XIV. To find the Time of the Mean Syzygies in any given Year and Month. 

From the Tables of the mean motion of the moon from the sun take out the motions for the given 
year and month ; add them, and subtract the sum from 12 signs ; then, if the time of mean conjunction 
or new moon be sought, the remainder, or the nearest less than it, being found in the Table of days, 
will give the day of mean new moon ; and after subtracting the number found in the Table, the re¬ 
mainder or the next less is to be found in the Table of hours, which will give the hour of the day when 
it is mean new moon ; and after another subtraction, the remainder is to be fouud in the Table of min¬ 
utes for the minute of mean new moon, and the next remainder will give the seconds. But if the time 
of mean opposition or full moon be sought, add six signs to the first remainder, and the sum being 
found in the Table of days, will give the day of mean full moon; the hours and minutes are to be found 
as before. 

Exam. 1 . Required the time of mean new moon in January, 1797. Ans. Jan. 27d. 7h. 40m. 21s. 

ft o / II 


1797 

Remains 
27 days 

7 hours 

40 minutes 


—0 26 57 10 
12 0 0 0 

11 3 2 50 

— 10 29 9 0 

3 53 50 
—3 33 20 

20 30 
—20 19 


21 seconds 


10 55 












246 


APPENDIX TO THE ASTRONOMY. 


In this operation the Tables give the moon’s mean distance from the %un on the 1st day of January 
Os. 26° 57' 10", which is subtracted from 12 signs to find how much the moon wants of a conjunction 
with the sun ; the remainder shows that she wants 11s. 3° 2' 50". In the Table of days it is found that 
she moves over 10s. 29° 9' in 27 days; therefore the mean new moon happens on the 27th day of 
January. 

Subtract 10s. 29° 9' from 11s. 3° 2' 50", there remains 3° 53' 50'", which in the Table of hours gives 
7 hours ; and subtracting 3° 33' 20" there remains 20' 30" which gives 40'; and by subtracting again 
there remains 10" 55"', and this gives 21 seconds. 

2. It is required to find the time of mean full moon in January 1797. Jins. Jan. 12d. 13h. 18m. 18s. 



S ° 

/ 

// 

1797 

—0 26 

57 

10 


12 0 

0 

0 


11 3 

2 

50 

Add 

6 0 

0 

0 


5' 3 

2 

50 

12 days 

—4 26 

17 

20 


6 

45 

30 

13 hours 

—6 

36 

12 



9 

18 

18 minutes 


—9 

8 

18 seconds 



9 


By this pVoblem the times of mean new and full moon may be calculated for every month in the 
year, or any longer time, with very little trouble ; for having found the time of mean conjunction and 
opposition in the month of January, add to these times a mean lunation, viz. 29d. 12h. 44m, 3s. contin¬ 
ually, rejecting the days in the month wherein the mean new or full moon is required, and the times 
will be known. 

Exam. Let it be required to find the mean new and full moons in every month of the year 1797. 


Mean full Moon. 

Mean new moon. 

D. h. m. s. 
January 12 13 18 1 

One lunation+29 12 44 3 

D. h. m. s. 
January 27 7 40 21 

One lunation + 29 12 44 3 

February 11 2 2 21 

February 25 20 24 24 

March 12 14 46 24 

March 27 9 8 27 

April 11 3 30 27 

April 25 21 52 30 

May 10 16 14 30 

May 25 10 36 33 

June 9 4 58 33 

June 23 23 20 36 

July 8 17 42 36 

July 23 12 4 39 

August 7 6 26 39 

\ugust 22 0 48 42 

September 5 19 10 42 

September 20 13 32 45 

October 5 7 54 45 

October 20 2 16 48 

November 3 20 38 48 

November 18 15 0 51 

December 3 9 22 51 

December 18 3 44 54 


JYote. The mean and true syzygies never happen at the same time, except when the moon is in 
or very near her apogee or perigee ; for in these points many of her inequalities either vanish or 
are very small; when the moon is in any other point of her orbit there is some interval of time be¬ 
tween the mean and true conjunctions or oppositions. The greatest is about 14 hours. 

































247 


USE AND APPLICATION OF THE TABLES, &c. 


PROB. X\ r . The Time of Mean Conjunction or Mew Moon being given , to find the True Time. 

Calculate the longitudes of both sun and moon for the time of mean conjunction (Prob. II.—VII.) ; 
and it they are equal to one another, the mean and true conjunctions happen at the same time ; but if 
they differ, subtract the least from the greatest. Find the hourly motions of both sun and moon at the 
time (Prob. IV.—XII.), and subtract the sun’s hourly motion from the moon’s, the remainder is the 
hourly motion of the moon from the sun ; and then, f 

As the moon’s hourly motion from the sun 
Is to one hour or 60 minutes ; 

So is the difference between the sun’s and moon’s long. 

To the time between the mean and true conjunction. 

If the moon’s longitude be less than the sun’s, the interval is to be added to the time of mean con¬ 
junction ; but if it be greater, subtract the interval; the sum or remainder is the time of true con¬ 
junction. 

When it is required to find the time of true conjunction very accurately, the sun’s and moon’s longitude 
must be calculated again for the time found; and their difference, if there be -any, turned into time, and 
applied to the time last found, will give the true time of conjunction. 

It the difference be great, it will be necessary to renew the operation again. 

Exam. It is required to find the time of true conjunction in January 1797 ; the time of mean con¬ 
junction or new moon being January 27th, 7h. 40m. 21s. in the meridian of Greenwich. Ans. Jan. 
27d. 13h. 40m. 


The sun’s longitude then is 10s. 8° 28' 38", and his mean anomaly 6s. 28° 3' 51". 
For the Moon's Longitude. 


m. Long. 

Anomaly. 

£1 

S 0 ' " 

1797 10 7 34 43 

Jan. 27th 11 25 45 46 

7h. 3 50 35 

40 m. 21 58 

21s. 11 

S ° ' " 

0 24 6 53 

11 22 45 17 
3 48 38 
21 47 
11 

S 0 ' " 

3 1 15 6 

— 1 25 47 
56 
5 

10 7 33 13 
10 equations + 25 41 

0 21 2 46 

— 10 25 

2 29 48 18 

a or. 

2 29 43 53 

= 11th arg. 

10 7 58 54 

Equat. of centre —2 9 10 

0 20 52" 21 

-f- 25 41 

10 5 49 44 

Variation —3 14 

0 21 18 2' 

10 5 46 30 

13th + 1 5 


10 5 47 35 

©’s long. 10 8 28 38 

Short of conj. 2 41 3 


Moon’s hourly motion 
Sun’s ditto 


Arguments. 

Equations. 


S 0 ' " 

+ 

— 

1 

6 28 3 51 

' // 

5'21" 

2 

6 26 13 1 

0 25 


3 

5 0 5 19 


0 38 

4 

0 19 11 56 

0 19 


5 

11 7 6 24 

30 53 


6 

6 5 10 15 


0 11 

7 

4 8 • 2 33 

0 36 


8 

5 22 58 55 

0 5 


9 

11 8 1 49 

0 31 


10 

4 21 19 40 


0 58 

12 

11 27 21 6 

+ 32 49 

7 8 

13 

1 20 47 12 

— 7 8 




+ 25 41 



29 47 
2 33 


<- 


Hourly motion } from © 
As 27' 14" 

To lh. 

So 2° 41' 3" 

To 5h. 54m. 49s. 

Mean conjunction January 
Interval add 

Ans. New Moon January 


27 14 


D. h. m. s. 
27 7 40 21 

-f- 5 54 49 


27 13 35 10 































248 


APPENDIX TO THE ASTRONOMY. 


The time of true opposition or full moon is found in the same manrrer as the time of conjunction. 
The operation is to be continued until the difference of the sun and moon’s longitude be 6 signs. 

The rules here given for calculating the times of new and full moon are always good ; but the tedi¬ 
ousness of the operation has given occasion to the invention of other Tables, whereby the problem 
may be solved with less trouble. The best of these were published among Dr. Halley’s Astronomical 
Tables in the year 1749, but not as his invention. The ingenious Mr. James Ferguson having adapted 
them to the Gregorian style, and made some alterations in the arrangement, printed them in his As¬ 
tronomy about the year 17C0. Such of them as were judged proper for this work, are to be found in 
the Tables of mean new moons, &,c. extended to the year 1821. 

These Tables contain the times of mean conjunction or new moon in March for every year, which is 
made the first month with the mean anomalies of the sun and moon, and also the sun’s mean distance 
from the moon’s node at the time, with equations for reducing the mean to the true time of conjunction 
or opposition, whereby these times may be found for any month of a given year within the limits of the 
Table. 

PROB. XVI. To calculate the Time of New or Full Moon for any Month of a given Year by the Tables. 

1st. For the Month of March. 

Write down the time of mean new moon in March for the given year, with the mean anomalies of 
the sun and moon, and also the sun’s mean distance from the moon’s ascending node, out of the Table. 

When the time of Full Moon in March is required. 

If the new moon happens before the 15th of the month, add half a lunation, with the anomalies, &c. 
to the former numbers for new moon, the sum is the time of full moon ; but if it happens after the 15th, 
subtract half a lunation with the anomalies from the numbers for new moon, and the time of mean full 
moon in March will be known. 

2d. For any Month after March. 

When the time of mean new or full moon is required in any month after March, take out the numbers 
for March as before, and under them write down as many lunations with their anomalies as the given 
month is after March; and by the sum of these, the time of mean new or full moon may be known, 
together with the mean anomalies and the sun’s distance from the moon’s node ; which are the argu¬ 
ments for finding the several equations, to reduce the time of the mean syzygies to the true. 

With the sum of the days enter the Table of days, under the given month; and opposite to that 
number in the left hand column is the day of the mean syzygies ; but if the sum be less than any of 
those under the given month, add a lunation with the anomalies to the former numbers, and then enter 
the Table with the sum under the given month, and in the left hand column is the day of the month 
requited. 

The time of mean syzygy being known, to find the true by the Tables. 

1. With the sun’s mean anomaly enter the Table, and take out the first equation (making proportion 
for the odd minutes, &c.), and apply it to the time of the mean syzygy according to its sign. 

2. With the sun’s mean anomaly take the equation of the moon’s mean anomaly out of the Table, 
and apply it according to its sign, and the moon’s equated anomaly will be known. 

3. With the moon’s equated anomaly take the second equation out of the Table, which being applied 
to the former time, according to its sign, the result will be the time of the syzygy very nearly. 

4. Subtract the moon’s equated anomaly from the sun’s mean anomaly, and with the remainder take 
the third equation out of the Table, and apply it to the former equated time according to its sign. 

Lastly, with the sun’s mean distance from the moon’s node take the fourth equation out of the Table, 
and apply it to the last found equated time, according to its sign, and the result is the time of the true 
syzygy. 


USE AND APPLICATION OF THE TABLES, &c. 

Exam, 1 . Required the time of full moon in January, 1797. 



© 

’s anom 

. 

D 

’s anom. 

0’s 

dist 

. from ft,. 


D. 

h. 

m. 

s. 

S. 

O 

/ 

n 

S. 

O 

/ 

n 

S. 

O 

' 


1796, March 

8 

11 

39 

44 

8 

8 

3 

47 

3 

7 

2 

14 

8 

0 

26 

47 

i lunation 

+ 14 

18 

22 

2 

0 

14 

33 

10 

6 

12 

54 

30 

0 

15 

20 

7 

Full in March 

23 

6 

1 

46 

8 

22 

36 

57 

9 

19 

56 

44 

8 

15 

46 

54 

Add 10 lunations 

295 

7 

20 

30 

9 

21 

3 

14 

8 

18 

10 

4 

10 

6 

42 

20 

1797, January 

12 

13 

22 

16 

6 

13 

40 

11 

6 

8 

6 

48 

6 

22 

29 

14 

1st equation 

+ 

1 

0 

30 

—6 

8 

29 

38 


+ 

22 

50 

arg. 

4th 

equat. 

\ 

12 

14 

22 

46 

0 

5 

10 

33 

6 

8 

29 

38 





2d equation 


- 1 

20 

45 

arg. 

3d. 

equat. 

arg. 

2d. 

equat. 






12 

13 

2 

1 













3d equation 




26 













4th equation 


+ 1 

6 













A ns. January 

12 

13 

2 

41 














2. It is required to find the time of conjunction, or new moon, in the month of August, 1820. 



O’s Anomaly. 

Vs 

4nomaly. 

O’s dist. from 

D’s ft. 


D. 

h. 

m. 

s. 

S. 

0 

t 

/' 

d- 

O 

/ 

// 

S. 

O 

/ 

n 

1820, March 

14 

1 

42 

45 

8 

12 

25 

31 

6 

24 

37 

23 

11 

19 

35 

34 

5th luna. 

147 

15 

40 

15 

4 

25 

31 

37 

4 

9 

5 

2 

5 

3 

21 

10 

August 

8 

17 

23 

0 

1 

7 

57 

8 

11 

3 

42 

25 

4 

22 

56 

44 

1st equat. 

— 

- 2 

31 

47 

-11 

2 

44 

51 



57 

34 

arg. 

4th. 

equat. 


8 

14 

51 

13 

2 

5 

12 

17 

11 

2 

44 

51 





2d equat. 


- 4 

47 

7 

arg. 

3d. 

equat. 

arg. 

2d. 

equat. 






8 

10 

4 

6 













3d equat. 



- 4 

27 













4th 



- 1 

29 













Ans. Aug. 

8 

9 

58 

10 














Note. 1. These examples are wrought for the meridian of Greenwich; but the answers may be re¬ 
duced to the time under any other meridian by adding or subtracting the difference of longitude in time 
to or from them, according as the place is situated to the east or west of Greenwich. 

2. These Tables give the times of new and full moon with little trouble, and sufficiently true for 
common use; being rarely above 1 or 2 minutes wide of the truth: but when it is required to find 
the moment of conjunction or opposition accurately, calculate the longitudes of the sun and moon for 
the time found by the Tables; and if they are the same or differ 6 signs, according as the new or full 
moon is required, the time is truly found; but if not, take their difference. Find the hourly motion 
of the moon from the sun ; and then, 

As the hourly motion of the moon from the sun 

Is to 60 minutes ; 

So is the difference between the sun and moon’s long. 

To a number of minutes, &.c. 

which, applied ,to the time formerly found, will give the true time. 

3. The precise moment of conjunction or opposition is seldom necessary except in calculating 
eclipses. 

4. In the preceding examples the time of orbit conjunction or opposition is taken for the answer; 
that is, the time when the moon’s longitude in her orbit is the same with the sun’s longitude in the 
ecliptic, is taken for the time of conjunction ; and the time when the moon’s orbit longitude differs 6 
signs from the sun’s longitude, is taken for the time of opposition. 

* The answers are in mean time; and if the equation of time be applied to them with a contrary sign 
to that in the Table, the apparent time will be known. 

32 

























250 


APPENDIX TO THE ASTRONOMY. 


To calculate more accurately the time of new moon in August, 1820: The answer by our Tables 
is August 8d. 9h. 58m. 10s. The sun’s longitude at this time is 4s. 16° 9' 44", and his mean anomaly 
Is. 7° 23' 13". 


The calculation of the moon’s longitude for the same time follows. 


3 ’s m. Long. 

3’s anom. 

v«a- 

S. • ' " 

1820 3 19 18 47 

August 8th 0 18 48 26 

9h. 4 56 28 

58m. 31 50 

10s. 5 

11 0 1 6 
11 24 17 51 
4 53 58 
31 34 
5 

S. ° ' T. 

0 6 25 42 

— 11 39 0 

— 1 12 
7 

4 13 35 36 

10 equations — 25 26 

10 29 44 34 
+ 12 54 

11 24 45 23 
+ 5 28 

4 13 10 10 
Equat. centre +314 

10 2 *1 57 28 
— 25 26 

11 24 50 51 

4 16 11 14 
Variation + 1 

10 29 32 2 

arg. 11th. 

4 16 11 15 
13th equation — 1 1 


3’s orb. long. 4 16 10 14 
Sun’s long. 4 16 9 44 

Difference 30 


Arguments. 


1 

2 

3 

4 

5 

6 

7 

8 
9 

10 

12 

1 






+ 


S. 

0 

/ 

II 

/ 

n 

1 

7 

23 

13 

6 

37 

1 

2 

14 

57 



10 

17 

28 

31 

0 

51 

10 

24 

36 

18 



0 

25 

7 

10 



2 

2 

30 

23 

1 

49 

11 

17 

43 

57 



9 

22 

21 

21 



0 

2 

40 

29 



7 

8 

35 

39 

0 

57 

0 

0 

1 

30 

+ 10 

14 

10 

13 

8 

46 




Equations. 


0 29 

0 36 
33 43 

0 10 
0 39 
0 3 


35 40 
10 14 


25 26 


Moon’s hourly motion 28 54 
Sun’s ditto. 2 24 


Hourly motion 3 from © 26 30 
L. L. 

As 26' 30" 3549 

To 60m. 0 

So is 30" 20792 


Tolm. 7s. 17243 

D. h. m. s. 

Former time August 8 9 58 10 
Interval subtract — 1 7 


True time of new moon 8 9 57 3, which is nearly the same with the former. 

PROB. XVII. To find out the number of Eclipses there are in any given Year , and in what Months and 

Days they happen. 

From the Table of the moon’s mean motion take out the mean longitude of the moon’s nodes for the 
given year; and then by the Table of the sun’s longitude for every day of the year, find when the 
sun’s longitude wili be nearly the same with that of the nodes, for at these times the eclipses must hap¬ 
pen. The months when the sun is in or near the places of the nodes may be called the node-months. 
In this inquiry it may be proper to remember that the moon’s nodes move backward about 1° 38' every 
month. 

To find the Days when Eclipses happen. 

Calculate the times of the mean syzygies in the node-months out of the Tables, and also the sun’s 
distance from the moon’s node ; and if the sun’s distance from the node at the time of new moon be less 
than 18 degrees, an eclipse of the sun may be expected ; . and if the sun’s distance from the node at 
the time of full moon be less than 12 degrees, there may be an eclipse of the moon. We say there mav 
be, because we speak only of mean motions here. 


























251 


USE AND APPLICATION OF THE TABLES, kc. 

Exam. It is required to find the number of eclipses in the year 1796 ; and on what days they happen. 
The mean longitude of the moon’s ascending node, on the 1 st of January 1796, is 3s. 20° 35', and ot 
the descending node 9s. 20 ° 35'. The sun enters the sign Cancer about the 20 th of June, and the sign 
Capricorn about the 22d of December; therefore the node-months are January, July, and December. 


1 . Calculate the time of mean 

New moon. 

new moon in January. 

O’s dist. from j) ’s 

■ D. h. m. s. 

1795, March 20 2 51 8 

10 lunations -f- 295 7 20 30 

S. ° ' ", 

7 22 24 0 

10 6 42 20 

New D 1796 Jan. 9 10 11 38 

5 29 6 20 sun eclipsed. 


The sun’s mean distance from the moon’s ascending node being 5s. 29° 6 ' 20 ", his distance from the 
descending node is only 53' 40" ; and therefore the sun will be eclipsed to some place of the earth. 


2 . For the time of mean new moon in July. 


New moon. 


O 


1796, March 
4 lunations 


D. h. m. s 

8 11 39 44 
+ 118 2 56 12 


S, 

8 

4 


New 5 July 


4 14 35 56 


0 


’s dist. from D’s £1 

_ - // 

0 26 47 
2 40 56 


3 7 43 sun eclipsed. 


3. For the time of mean full moon in December. 


New Moon 

Sun’s dist. from & 

D. h m. s. 
1796, March 8 11 39 44 

i lunation + 14 18 22 2 

S. 0 ' " 

8 0 26 47 

+ 15 20 7 

Full D in March 23 6 1 46 

9 lunations -j- 265 18 36 27 

8 15 46 54 
+ 9 6 2 6 

Full D Dec. 14 0 38 13 

5 21 49 0 moon eclipsed. 


The sun’s distance from the moon’s ascending node being 5s. 21 ° 49', the moon’s distance from her 
descending node is only 8 ° 11 ', and therefore she is eclipsed. 

4. For the time of mean new moon in December. 

Add 1 lunation to the numbers for full moon. 


New 

moon. 




Sun’s dist. from SI. 


D. 

h. 

m. 

S. 

S. ° ' 

'/ 

Full D Dec. 

14 

0 

38 

13 

5 21 49 

0 

^ lunation 

+ 14 

18 

22 

2 

+ 15 20 

7 

New D Dec. 

28 

19 

0 

15 

6 7 9 

7 sun eclipsed. 


We have found that there are four eclipses in the year 1796, viz. three of the sun and one of the 
moon. The times of the mean conjunctions and oppositions are : 

D. h. m. s. 

! Jan. 8 10 11 38 

July 4 14 35 56 

Dec. 28 19 0 15 

And the moon is eclipsed - - —14 0 38 13 


PROB. XVIII. To calculate an Eclipse of the Moon. 

1 , Calculate the true time of opposition by Prob. XVI. and compute the longitude of the sun, and the 
orbit longitude of the moon, for that time ; if these differ 6 signs, the time of opposition is truly found ; 
but if their difference be less or greater than six signs, mark the defect or excess, and find the moon’s 
hourly motion from the sun ; then, as the hourly motion of the moon from the sun is to one hour, so is 
the defect or excess to the interval; which being applied to the time formerly found, gives the true 



















252 


APPENDIX TO THE ASTRONOMY. 


time of opposition. This operation must be repeated until the longitude of the sun and the orbit 
longitude of the moon differ precisely 6 signs. 

2 . Calculate the moon’s latitude, horizontal parallax, and hourly motion, by the Tables ; take also 
the sun and moon’s semidiameters out of the Tables. 

3. Add the sun’s horizontal parallax (which is 9 seconds) to the moon’s for the time, and from the 
sum subtract the sun’s semidiameter; the remainder is the semidiameter of the earth’s shadow; to 
which add the moon’s semidiameter, and the sum is the semidiameter of the moon and earth’s shadow ; 
from which subtract the moon’s latitude, and the remainder is called the parts deficient. If these be less 
than the moon's diameter, the eclipse will be partial; if they be equal to the moon’s diameter, the 
eclipse will be total without continuance ; and if they be greater than the moon’s diameter, the eclipse 
will be total wi f h continuance. 

Exam. 1 . It is required to calculate the eclipse of the moon, which will happen in December, 1797. 


1 . Calculate the time of opposition. 



Sun’s Anom. 

AToon’s Anom. 

0’s dist. from 

D. h. rn. s. 

1797, new j) March 27 9 12 24 

^ lunation — 14 18 22 2 

S. ° ’ " 

8 26 25 59 
— 14 33 10 

s. 0 7 " 

2 12 39 19 
— 6 12 54 30 

s. 0 ' " 

9 9 9 48 

— 15 20 7 

Full moon March 12 14 50 22 

9 lunations -}-265 18 36 27 

8 11 52 49 
8 21 56 54 

7 29 44 49 
7 22 21 4 

8 23 49 41 

9 6 2 6 

Full moon December 3 § 26 49 

1 equation — 1 52 47 

5 3 49 43 

— 3 21 23 13 

3 22 5 53 

— 42 40 

5 29 51 47 
arg. 4th equat. 
Moon eclipsed. 

3 7 34 2 

2 equation -f 8 53 8 

3 16 27 10 

3 equation — 3 11 

4 ditto v — 1 

Full moon December 3 16 23 58 

1 12 26 30 

arg 3d. equat. 

3 21 23 13 

arg. 2 d equat. 


2 . Calculate the sun and moon’s longitude for the time of full moon. 
For the Sun’s Longitude. 


Sun’s Long. 

Sun’s Anom. 

§7"° 7 TT 

1797 9 10 37 33 

December 3d 11 2 9 47 

16h. 39 25 

23m. 57 

58s. 2 

S. ° ' " 

6 1 8 17 

11 2 8 46 

39 25 

57 

2 

8 13 27 44 

5 3 57 27 

Equation — 51 45 


Sun’s longitude 8 12 35 59 





# 




























253 


USE AND APPLICATION OF THE TABLES, &c. 

For the Moon’s Longitude. 


Moon’s m. Long. 

Moon’s Ariom. 

Moon’s 

go/// 

1797 10 7 34 43 

Dec. 3d. 4 0 26 45 

16h. 8 47 3 

23m. 12 37 

58s. 32 

go / // 

0 24 6 53 

2 22 54 4 

8 42 36 
12 3i 
32 

S. 0 ' ", 

3 1 15 6 

— 17 50 45 
— 2 7 

— 3 

2 17 1 40 

10 equations + 1 25 1 

3 25 56 36 
+ 9 44 

2 13 22 11 
+ 43 

2 18 26 41 
Equat. centre. — 5 45 57 

3 26 6 20 

+ 1 25 1 

2 13 26 14 

2 12 40 44 
Variation + 6 

3 27 31 21 

arg. 11th. 

2 12 40 50 
13th equation — 113 


Moon’s orb. Ion. 2 12 39 37 
Sun’s Ion. 8 12 35 59 


Past opposition 3 38 


Arguments 

Equations. 







+ 





S. 

0 

/ 

// 

O 

/ 

// 

/ 

II 

, 

5 

3 

57 

27 

0 

5 

0 



2 

5 

12 

48 

49 




0 

17 

3 

7 

4 

53 

55 

, 0 

0 

43 



4 

4 

4 

47 

58 

0 

0 

47 



5 

8 

12 

54 

46 

1 

17 

15 



6 

1 

16 

52 

13 

0 

1 

30 



7 

3 

8 

57 

19 

0 

0 

46 



8 

10 

21 

59 

9 




0 

26 

9 

2 

8 

29 

5 




0 

19 

10 

6 

0 

46 

12 , 

0 

0 

2 



12 

6 

0 

4 

45 

+ 1 

26 

3 

— 1 

2 

13 

8 

0 

57 

51 

— 0 

1 

2 



14 

11 

29 

9 

29 











+ 1 

25 

1 




Moon’s hourly motion 
Sun’s ditto 


- 35 18 

2 32 


Hourly motion j from © 


- 32 46 

L. L. 


As hourly motion } from © 32' 46" 2627 
To one hour or 60m. 0 

So is 3' 38" - - - - - 12178 


To 6m. 39s. 

Former time December 


9551 


D. 

3 


Interval subtract 


. I 


h. ' " 

16 23 58 
— 6 39 


Time of full moon 


3 16 17 19 


Sun’s longitude 
Moon’s ditto 
Reduction 


For the Moon’s Latitude. 
Arguments. 


S. 


11 29 13 23 
0 0 53 23 
6 25 15 56 
2 29 19 20 
11 3 22 44 


— 4 11 
+ 8 
+ '6 

— 25 

— 7 


Moon’s lat. S. — 4 29 - 


8 12 35 43 
2 12 35 43 

+ 12 

For the Moon’s Horizontal Parallax. 


Arguments 


go/// 

1 

8 12 54 46 

2 

3 27 31 21 

3 

6 0 4 45 


+ 0 11 

+ 58 32 
+ 0 27 


Equatorial Parallax 
Reduction 


59 10 
— 9 


Horizontal Parallax 


59 1 








































APPENDIX TO THE ASTRONOMY. 


For the time of Reduction. 

As moon’s hourly motion from © 32' 46 
To lh. or 60'. 

So is reduction 12 " 


To time of reduction -f- 22" 

/ // 

Sun’s semidiameter 16 17 

Moon’s semidiameter 16 8 

Angle of the moon’s path with the ecliptic 5°40'. 

In eclipses, the disk of the sun or moon is supposed to be divided into twelve equal parts 
called digits, and each of these again divided into 60 equal parts or minutes. The magnitude of an 


eclipse is estimated by the number of digits. 

Calculation of the Eclipse. 

> H 

Moon’s horizontal parallax - - 59 1 

Sun’s ditto - - - * - - + 9 

Sum - - - - - - 59 10 

Sun’s semidiameter subtract - - —16 17 

Semidiameter of the earth’s shadow - 42 53 

Moon’s semidiameter add - - -f 16 8 

Semid. of the moon and earth’s shadow 59 1 

Moon’s latitude subtract - —4 29 

Remains the parts deficient - - 54 32 

The eclipse will be total. 


» 

L. L. 

2627 

0 

24771 


22144 


For the Digits eclipsed. 

L. L. 

As moon’s semidiameter 16' 8 " - - 5704 

To 6 digits . 10000 

So parts deficient 54' 32" - - - - 415 


To digits eclipsed 20 ° 16' 

Mean time of orbit opposition Dec. 
Time of reduction add and subtract 

Middle of the eclipse ... 

Ecliptic opposition - 

To each of these add the equation of time 

Appar. time of the middle of the eclipse 
-of ecliptic opposition 




4711 

D. 

h. 

/ 

// 

3 

16 

17 

19 



± 

22 

3 

16 

17 

41 

3 

16 

16 

57 


+ 

9 

22 

3 

16 

27 

3 

3 

16 

26 

19 


To find the Scruples of Incidence , or that Part of the Moon’s Path which she passes over between the begin¬ 
ning and middle of the Eclipse. 

Reduce the sum of the semidiameters of the moon and earth’s shadow to seconds, and also the moon’s 
latitude ; find their sum and difference ; and to the logarithm of their sum add the logarithm of their dif¬ 
ference, and divide the sum by 2 , the quotient is the logarithm of the scruples of incidence in seconds. 
In this example the semidiameter of the moon and earth’s shadow is 











255 


USE AND APPLICATION OF THE TABLES, &c. 


/ n // 

59 1 = 3541 

Moon’s latitude 4 29 = 269 Logarithms. 


Sum 3810 3.5809450 

Difference ' 3272 3.5148133 


2)7.0957583 


Scruples of incidence 3530 3.5478791 

or 58' 50" 

For the time of half duration. - 


• 

/ ff 


L. L 

As hourly mo. D from 0 

= 32 46 


2627 

To 1 hour or 60 minutes 



0 

So scruples of incidence 

58 50 

(29' 25") 
h. ' 

(0 53 51) 

2 

3096 

To time of half duration lh. 47' 42". 

469 



1 47 42' 



Because the terms of this proportion are such, that the fourth term comes out too great for the Table 
of logistical logarithms, take half of the third term ; and then double the fourth term for the answer. 

D. h. m. s. 

Apparent time of the middle of the eclipse 3 16 27 3 

Time of half duration, subtract and add db 1 47 42 


Beginning - - - - - - 3 14 39 21 

End.3 18 14 45 


In total eclipses of the moon, she continues some time in total darkness; to calculate this time, it is 
necessary to find the half length of that part of the moon’s path which lies wholly within the earth's 
shadow, which some call the scruples of total darkness. The operation is performed thus : 

Subtract the moon’s semidiameter from that of the earth’s shadow ; reduce the remainder, and also the 
moon’s latitude, to seconds; then find their sum and difference, and take the half sum of the logarithms 
of the sum and difference, and the number answering is the scruples of total darkness. 

In this example, 

7 If 

The semidiameter of the earth’s shadow is 42 53 

The moon’s diameter - - — 16 8 



/ 

ft ft 

26 45 

The difference is 

26 

45 = 1605 


Moon’s latitude 

4 

29 = 269 


Sum 


1874 

Logarithms. 

3.2727696 

Difference - 

- 

- 1336 

3.1258065 


2)6.3985761 


Scruples - 1584 3.1992880 

Or 26'24". 

For the time of half duration of total darkness. 

f // 

L. L. 

As hourly motion of moon from sun = 32 46 2627 

To one hour or 60 - - - 0 

So are scruples - - 26 24 3565 


To time of half duration 48' 21 " - - - 938 














256 


APPENDIX TO THE ASTRONOMY. 


D. h. m. 3. 

Apparent time of the middle of the eclipse 3 16 27 3 

Half duration of total darkness subtract and add 43 21 


Beginning of total darkness - - 3 15 38 42 

End of ditto - - - -31715 23 

From the calculation it appears, that the moon will be eclipsed at London, 1797, December 3d. 

Appar. time, 
h. ' 

14 39 

15 38 

16 26 

16 27 

17 15 

18 14 

Digits eclipsed 20° 16' from S. side of earth’s shadow. 

Duration of the eclipse 3h. 35'. 

Duration of total darkness lh. 36'. 


Beginning of the eclipse 
Beginning of total darkness 
Ecliptic opposition 
Middle of the eclipse 
End of total darkness - 
End of the eclipse 


Exam. 2 . Calculation of an eclipse of the moon which will happen 1802, Sept. 11th. 

Meantime of orbit opposition, Sept. lid. lOh. 40'. At that time sun’s longitude 5s. 18° 25' 16" 
moon’s longitude 11s. 18° 25' 26", latitude north 37' 6" ascending. Reduction = 1' 40", time of reduc 
tion 2' 48". 

i a 

Moon’s hourly motion - - - 38 4 

Sun’s ditto - - - — 2 26 


Moon’s hourly motion from sun - 35 38 

O / // 


Moon’s semidiameter 

- 

0 16 43 

Sun’s ditto - 


- 9 15 57 

Moon’s horizontal parallax 

- 

0 66 17 

Angle of the moon’s path with the ecliptic is 

5 38 0 

Calculation of the Eclipse. 

, / 

Moon’s horizontal parallax 

. 

61 17 

Sun’s ditto 


- + 9 

Sum - 

- 

61 26 

Sun’s semidiameter subtract 

- 

— 15 57 

Semidiameter of the earth’s shadow 


45 29 

Moon’s semidiameter, add 


+ 16 43 

f 

Semidiameter of moon and earth’s shadow 

_ 

62 12 

Moon’s latitude, subtract 


—37 6 

Parts deficient 

- 

25 6 

For the digits eclipsed. 

0 1 

n _ _ 

As moon’s semidiameter 

0 16 

L. L. 

43 5550 

To 6 digits - 

- 

10000 

So parts deficient - 

0 25 

0 3785 

To digits ^ 

9 0 

0 8236 


\ 









USE AND APPLICATION OF THE TABLES, kc. 


D. h. m. s. 


Mean time of orbit opposition 

- 

11 10 40 0 

Time of reduction, subtract and add 

- 

q: 2 48 

Middle of the eclipse 


11 10 37 12 

Ecliptic opposition 

- 

11 10 42 48 

Equation of time add to each 

- 

-f 3 31 

Apparent middle of the eclipse 

• 

11 10 40 43 

-ecliptic opposition 

- 

11 10 46 19 

For the scruples of incidence. 

*- 

Semid. of the 2 and ©’s shadow 62 

12 = 

3732 

Moon’s latitude - - 37 

6 = 

2226 

-- Logarithms. 

Sum 

Difference - 


5958 3A761O05 
1506 3.1778250 

2)6.9529255 

Scruples - 

Or 49' 45" 

For the time of half duration. 

/ " 

2905 3.4764627 

t. c. 

As moon’s hourly motion from sun 

35 38 

2263 

To 1 hour or 60m. 

_ 

0 

So i scruples of incid. 

9 59 

7789 

To of half duration 

16 48 
5 

5526 

Half duration - - lh. 24m. Os. 

D. b. m. s. 

Apparent middle of the eclipse 

- 

11 10 40 43 

Time of half duration, subt. and add 

- 

qp 1 24 0 


Beginning - - - - 11916 43 

End - - - 11 12 4 43 


1802, September 11th, the moon is eclipsed. 


Beginning - 

Middle - 

Ecliptic opposition 
End - 

Duration 2 hours 48 minutes. Digits eclipsed 9°. 


Appar. time, 
b. m. 

9 16 
10 40 
10 46 
12 4 


PROB. XIX. To describe a Figure representing a lunar Eclipse. (Plate XIV. Fig. 4.) 

Exam. 1 . It is required to project the eclipse, calculated in the 1st example of the last problem. 

1 . Draw the straight line AB for part of the ecliptic, in which take any point C, and draw CD at 
right angles to AB, for the axis of the ecliptic ; which is drawn downward, because the moon’s latitude 
at the beginning and middle of the ecliptic is south. 

2 . Take the semidiameter of the moon and earth’s shadow, 59' l'q from any scale, and with it, as a 
radius, describe the circle ADB; the moon’s centre is in one point of the circumference of this circle at 
the beginning ; and in the opposite point of it, at the end of a total eclipse. 

3. Take the semidiameter of the earth’s shadow, 42' 53", from the same scale, and with it, from the 
centre C, describe the circle KLM, to represent the earth’s shadow. 

4. Make CD the radius of a line of chords on the sector, and from it take the angle of the moon’* 
path with the ecliptic 5° 40', and set it from D to E toward the left hand, (because the moon’s latitude 
is south ascending northward) and draw the line CE for the axis of the moon’s orbit. 

33 












258 


APPENDIX TO THE ASTRONOMY. 


5. Take the moon’s latitude 4' 29" from the scale, which set from C to F on the line CE; and 
through F draw the straight line FN at right angles to CE; then the points N, F, and P are the places 
of the moon’s centre at the beginning, middle, and end of the eclipse. Take the moon’s semidiameter 
from the scale, and with it describe circles from the centres N, F, and P. These shall represent the 
moon at these points. 

To mark the Hours on the Moon's Path. 

The middle of the eclipse is December 3d. I6h. 27m. 3s. which, in civil reckoning, is 4h. 27m. in the 
morning of December 4th, marked on the moon’s path by the point F. Now, for the place of four 
hours, say, 

As one hour 

Is to the moon’s hourly motion from the sun 32' 46"; 

So is 27m. 3s. 

To 14' 46", the distance of 4h. from F. 

Take 14' 46" from the scale, and set it from F, toward the right hand for the place of 4 hours; then 
take the moon’s hourly motion from the sun 32' 46" and set it on the moon’s path from 4 to 3, and to 5, 
and also from 5 to 6, and the hours are marked. Divide each of the hour distances into six equal parts, 
and the place of the moon’s centre will be known at every 10 minutes. 

Exam. 2 . Required the projection of the eclipse calculated in the 2d example of the last problem. 
(Plate XIV. Fig. 5.) 

This is done in the same manner as the former; only, because the moon’s latitude is north, the semi¬ 
circles ADB and KLM must be above the straight line AB ; and because the moon’s latitude is ascend¬ 
ing northward, the angle of her path with the ecliptic 5° 38' must be set from D to E toward the right 
hand. Draw CE for the axis of the moon’s orbit, and thereon set the moon’s latitude 37' 6" from C to 
F ; and through F draw the straight line NP at right angles to CE ; and then the points N, F, and P, are 
the places of the moon’s centre at the beginning, middle, and end of the eclipse, as in the former example. 

Take the moon’s semidiameter 16' 43" from the scale, and with it, for a radius, describe circles on 
the centres N, F, and and P, which will represent the moon’s disk when she is at these points ; and then 
place the hours on the moon’s path, as directed in the former example. 

Previous to the Calculation and Projection of Solar Eclipses, it will be proper to premise some 
things for explaining the terms used in the graphical delineation of them. 

Although the earth is nearly a globe, yet when it is viewed from a very distant point, such as the 
centre of the sun, it will appear as a flat, round surface, like the moon, to us ; and this appearance is 
called the earth's disk. , 

By the continual rotation of the earth round its axis, every point on its surface describes a circle par¬ 
allel to the equator ; and the circle described by any point or place is called the path of the vertex of that 
place. 

The representations of the paths of places on the earth’s disk are not always of the same form, but 
differ with the sun’s longitude; for when the sun is in either of the equinoctial points, the paths of all 
places are represented by straight lines; .and when the sun is in any other point of the ecliptic, they 
are represented by ellipses, more or less eccentric according as the sun is nearer to, or farther from, 
the equinoctial circle. 

In describing these paths on the earth’s disk, respect must be had to the position of the axes of the 
earth and ecliptic at the time. 

When the sun is in any of the ascending signs V?, 3£, V, B, n, the northern half of the earth’s 

axis lies to the right hand of the axis of the ecliptic, as seen from the sun ; but when he is in any of the 
descending signs gz, SI, ^=, rq,, J , it lies to the left hand. When the sun is in either of the solsti¬ 
tial points, the two axes coincide ; and when he is in either of the equinoctial points, they form the 
greatest angle. 

When the sun is in any of the northern signs, Aries, Taurus, &c. the north pole of the earth is in the 
upper or enlightened part of the earth’s disk; and when he is in any of the southern signs, Libra, 
Scorpio, &,c. the north pole of the earth is in the under or obscure part of the disk. 

The transverse diameter of any path is always at right angles to the earth’s axis, and the conjugate 
diameter coincides with or is a part of it. 

PROB. XX. The Latitude of a Place , the Time of the Year , and the Sun's Longitude , being given , to de¬ 
scribe the Earth's Disk , and the Path of that Place thereon. 

Exam. It is required to describe the earth’s disk, and the path of a place in latitude 51° 32' N. 1803, 
August 17th, at 8h. 19m. in the morning, apparent time. (Plate XV. Fig. 2.) 



USE AND APPLICATION OF THE TABLES, &c. 

Sun’s longitude 4s. 23° 24' 45", distance from the solstice 53° 24' 45"; declination N. 13° 43' 43 . 

Draw the straight line AB to represent a part of the ecliptic, and let C be the point therein, which 
is opposite to the sun at the time: from C draw CH at right angles to AB, and CH is the axis of the 
ecliptic, and H its pole. 

From the centre C describe the semicircle AHB to represent the northern half of the earth’s disk. 
Make CA or CB the radius of the line of chords on the sector, and take the chord of 23 ^ degrees, 
which set from H both ways to f and g, and draw the straight line fVg’, the north pole of the earth is 
always in this line. 

Make /V the radius of the line of sines on the sector, and take the sine of 53° 24' 45", the sun’s dis¬ 
tance from the solstice, which set off - from V to P toward the left hand, because the sun is in the sign 
Virgo, and draw the straight line CP h for the earth’s axis; then P is the north pole of the earth. Or 
the angle contained between the earth’s axis and that of the ecliptic may be found more accurately by 
calculation, thus; 

O / 

As radius - - - - - s. yo o 10 . 

Is to thp sine of the sun’s dist. from the solstice - 53 25 9.9047106 

So is the tan. of the distance of the poles 23 28 9.6376106 


To the tan. of the angle contained by the axes 19 13 9.5423212 

Now set off the chord of 19° 13' from H to /t, and join CH, which will cut f g in P, the place of the 
north pole. 

Make CA the radius of the line of chords on the sector ; take the chord of 38° 28', the complement of 
the latitude, and set it off from A, both ways, to e and n, where make marks. Again, take the chord of 
the sun’s declination 13° 43' 43" from the sector, and set it off from the points e and », both ways, to 
D and F, and to M and G; then draw the lines DM, FG, cutting CP in 12 and 12 ; and the line 12 K 12 
is the conjugate axis of the ellipse, which bisect in K, and through K draw the line 6 K 6 at right 
angles to CP. 

Make CA the radius of the line of sines on the sector, and take the sine of the co-latitude 38° 28', and 
set it off from K, both ways, to 6 and 6. These hours fall on the circumference of the disk, when the 
sun is in either of the equinoctial points; but at all other points they fall within it. The line 6 K 6 is 
the transverse axis of the ellipse. 

Make K 6 the radius of the line of sines on the sector, and from it take the sines of 15°, 30°, 45°, 60°, 
and 75°, and set them off from K, both ways, toward 6 and 6; and through these points draw lines paral¬ 
lel to CP. Again, make K 12 the radius of the line of sines on the sector, and take from it the sines 
of 75°, 60°, 45°, 30°, and 15°, and set them off on the parallels, beginning with that next to 12 K 12, 
and through these points draw a curve line which will be the path of the given place. Mark the 
hours on the path as per Fig. and divide them into halves, quarters, and smaller parts, at pleasure. 

The 12 next to P marks midnight, when the sun’s declination is north; and that part of the path is 
below the disk; but the other 12 between K and C is noon or mid-day. The path touches the circum¬ 
ference of the disk a little before 5, which is the time of sun-rising; and again a little after 7, the time 
of sun-setting. 

In north latitude, when the sun’s declination is south, the 12 nearest to P is mid-day, and the other is 
midnight. 

In the same manner, the parallel of latitude, or path of any place, may be described on the earth’s 
disk for any given time, and is used in representing the appearance of an eclipse of the sun. 

PROB. XXI. To calculate an Eclipse of the Sun. 

1 . Calculate the time of mean conjunction, with the sun’s distance from the moon’s node, and thereby 
find if an eclipse will happen; and if it do, calculate the true time of conjunction, and the sun and 
moon’s longitudes at that time. 

2 . Calculate the moon’s latitude, horizontal parallax, horary motion, semidiameter, and angle of her 
path with the ecliptic. 

3 . Calculate the sun’s declination, hourly motion, and his distance from the solstice. 

Having found these necessary requisites, write them down in order to be ready for use. 

Exam. The sun will be eclipsed in August, 1803. 

The time of mean conjunction is Aug. 17d. 7h. 12' 24". 

Sun’s mean distance from the moon’s node 6s. 2° 8' 8". 

And the true conjunction is Aug. 16d. 20h. 30' 34" mean time. 


259 


i 



I 


I 


I 


260 


APPENDIX TO THE ASTRONOMY. 


f , 

S. 

O f ! 

The sun’s longitude then 

4 23 25 3 

The moon’s orbit longitude 

4 23 28 49 

Past conjunction 

0 

0 3 46 

Sun’s mean anomaly Is. 15° 

8' 15". 


Moon’s hourly motion 


30 57 

Sun’s ditto subtract 


2 24 

Moon’s hourly motion from sun 


28 33 



L. L. 

As moon’s hourly motion from © 28' 33' 

3225 

To 1 hour or 60 minutes 


0 

So is difference of longitudes 3' 46" 


12022 

To interval —7' 54" 


8796 


D. h. 

r n 

From the time formerly found 

16 20 

30 34 

Subtract the interval 


- 7 54 

Correct time of conjunction 

16 20 

22 40 

Equation of time 


- 3 48 

Apparent time of conjunction 

16 20 

18 54 


or 8h. 18' 54'' in the morning of August 17th. 


For the Sun and Moon’s places at the corrected time of Conjunction, 
m. ' " m. s. " 

As 60 : 2 24 :: 7 54 : 18 to be subtracted from ©’s long. 

60 : 30 57 : : 7 54 : 4' 4" to be subtracted from the 5’s long. 


S. 

And then the sun’s longitude is 4 

The moon’s ditto 4 

2. The moon’s horizontal parallax, or semi- > 

diameter of the earth’s disk \ 

3. Sun’s distance from the nearest solstice 

4. Sun’s declination north 

5. Angle of the moon’s path with the ecliptic 

6. The moon’s latitude south descending 

7. The moon’s hourly motion from the sun 

8. The sun’s semidiameter 15' 52" 

9. The moon’s semidiameter 15 14 

10. The semidiameter of the penumbra, or sum 
of the semidiameters of sun and moon 


O 

/ 

// 

23 

24 

45 

23 

24 

45 

0 

s 

o 

0 

55 

9 

53 

24 

45 

13 

43 

43 

5 

43 

0 

0 

0 

46 

0 

28 

33 

l o 

30 

56 


In this example the sun’s distance from the nearest solstice is found by subtracting 3 signs from his 
longitude. The angle of the moon’s path with the ecliptic is taken out of the Table. 

The method of finding the other numbers has been taught formerly. 


PROB. XXII. To describe a Figure representing a Solar Eclipse , and to find its Magnitude Beoinnino- 

Middle , and End at London. (Plate XV. Fig. 2.) 

Exam. Project the solar eclipse, calculated in the last problem. 

Draw the straight line AB to represent a part of the ecliptic; take the semidiameter of the earth’s 
disk from a scale, and with it as radius describe the semicircle AHB for the northern half of the earth’s 
disk, and thereon describe the path of London, as has been taught in the preceding problem. 

Make the semidiameter of the disk CA the radius of the line of chords on the sector, and take the 
chord of 5° 43' the angle of the moon’s path, which set off on the circumference of the disk, from 
H toward the right hand, to M, because the moon’s latitude is south descending, and draw CM for the 
axis of the moon’s path. 










261 


USE AND APPLICATION OF THE TABLES, &c. 

Set off the moon’s latitude 46" south on MC, produced below AB to X ; and through X, at right an¬ 
gles to MC, draw the line of the moon’s path, which cuts AB on the left hand of X. The point X is the 
time of conjunction by the Tables. 

To find the place of 8 hours on the moon’s path. 

L. L. 

As 1 hour or 60 minutes 0 

To the D’s hourly motion from the © 28' 33" 3225 

So is the time of conjunction after 8h. 18' 52' 5025 

To the distance of 8 hours from X 8' 58" 8250 

Take 8' 58" from the scale, and set it off from X toward the left hand for the place of 8 hours on the 
moon’s path. 

Take also from the scale 28' 33", the moon’s hourly motion from the sun, and set it off both ways 
from 8 to 9 and 7, and also from 7 to 6, &c. and the places of the hours on the moon’s path will be 
known. Divide each hour distance into 12 equal parts by dots, and the place of the moon’s centre, or 
rather the centre of the penumbra, will be known at every 5 minutes during the eclipse. 

To find the middle of the Eclipse. 

Apply one side of a square to the path of the moon or penumbra’s centre, and move it backward and 
forward until the other side of the square cuts the same hour and minute on the path of London and of 
the penumbra’s centre, which in this example is at y and u in 6 hours 32 minutes, the middle of the eclipse. 

Take the sun’s semidiameter 15' 52" from the scale, and with it describe a circle about the point y for 
the sun’s disk ; then take the moon’s semidiameter 15' 4" from the scale, and with it describe a circle 
about the point u for the moon’s disk at the middle of the eclipse. The part of the sun’s disk cut 
oil’ by the moon’s is the magnitude of the eclipse as it will appear at London. In this example it is 
about 3^ digits. 

Lastly, take the semidiameter of the penumbra 30' 56" from the scale; and setting one foot of the 
compasses on the moon’s path, and the other on the path of London, toward the left hand, carry that 
extent backward and forward until both points fall on the same hour and minute in each path, and that is 
the beginning of the eclipse at London. With the same extent of the compasses, and one foot on each 
path, carry them backward and forward toward the right hand ; and where both points fall on the same 
time, that will be the end of the eclipse at London. These trials give, 

h. m. 

The beginning of the eclipse at - - 5 50 

The middle - - - - - - 6 32 

The end.7 30 

Duration lh. 40m. Digits 3° 30'. 

Note*\. The projection of a solar eclipse will exhibit the appearance of it more naturally, if some 
alterations be made in the preceding process, adapted to the supposition, that the observer is on the earth. 

/ These alterations consist in drawing the axes of the earth and moon’s path on the side of the axis of 

the ecliptic contrary to that which is required by the rule, and in numbering the hours in the opposite 
direction. For the relative position of the axes, as seen from the sun, is inverted with respect to an 
observer of the sun on the earth. If a projection, made according to the preceding rule, be turned over 
from right to left, it will appear, if visible through the paper, to correspond with one of this construction. 
‘ Plate XV. Fig. 3, is an example, being the same eclipse projected in this manner, and requires no 
farther explanation. 

Note, 2. The situation of the point on the sun’s limb, with respect to a vertical and a horizontal 
diameter, where an eclipse begins, may be easily determined by Projection. Thus, with the points in the 
Respective paths, where the centres of the sun and moon are at the beginning of the eclipse, as centres, 
and their semidiameters as radii, describe circles, touching each other. Draw a line from the centre 
C (Fig. 2 or 3, Plate XV.) through the centre of the sun, and it will give the vertical diameter; and a 
diameter perpendicular to this will be horizontal. Then the line of chords on a sector being adapted to 
the seuiidiameter of the sun, the arc of the circumference, contained between the point of contact and 
an extremity of one of these diameters, measured on the chord line, will give the required situation ot 
the point, where the eclipse begins. 


s 



I 



SOLAR AND LUNAR TABLES, 


TABLES OF THE SUN’S MEAN MOTIONS. 


I. 

TQ 

Sun’s Mean Longitude and 

[1. 

The Pre- 


III. Sun’s Mean Longitude and ( 

Obliquity of the 


Anomaly in Julian Years. 


cession of the 


Anomaly in Years current. 

Ecliptic 1st 







Equinoctial 






January. 

Years. 

M. Longitude. 

M. Anomaly 


Faints. 


Yean. 

M» Longitude. 

M. Anomaly. 




S. ° ' " 

s. 

O / 

// 

O 

/ n 


A. D. i 

3. ° ' " ! 

3. 

o / II 

o / n 


1 

11 29 45 40 

11 

29 44 

35 

0 

0 50.3 


1761 < 

) 10 20 51 ( 


1 31 11 

23 28 16 


2 

11 29 31 21 

11 

29 29 

0 

0 

1 40.7 


1781 < 

3 10 30 71 

3 

1 18 27 

23 28 9 


3 

11 29 17 2 

11 

29 13 

44 

0 

2 31 


1791 ! 

d 10 5 12i 

S 

0 42 32 

23 27 50 

B 

4 

0 0 1 51 

11 

29 57 

27 

0 

3 21.4 


B 1792! 

3 10 50 1 i 

6 

1 26 15 

23 27 48 


5 

11 29 47 32 

11 

29 42 

2 

0 

4 11.7 


1793! 

3 10 35 41 

6 

1 10 49 

23 27 47 


6 

11 29 33 13 

11 

29 26 

37 

0 

5 2.1 


1794! 

9 10 21 22 

6 

0 55 24 

23 27 48 


7 

11 29 18 54 

11 

29 11 

12 

0 

5 52.4 


1795! 

9 10 7 3 

6 

0 39 59 

23 27 50 

B 

8 

0 0 3 43 

11 

29 54 

55 

0 

6 42.8 


B 1796 

9 10 51 52 

6 

1 23 42 

23 27 52 


9 

11 29 49 23 

11 

29 39 

29 

0 

7 33.1 


1797 

9 10 37 33 

6 

1 8 17 

23 27 54 


10 

11 29 35 4 

11 

29 24 

4 

0 

8 23.5 


1798 

9 10 23 13 

6 

0 52 51 

23 27 55 


11 

11 29 20 45 

11 

29 8 

39 

0 

9 13.8 


1799 

9 10 8 54 

6 

0 37 26 

23 27 58 

B 

12 

0 0 5 34 

11 

29 52 

22 

0 

10 4.2 


1800 

9 9 54 35 

6 

0 22 1 

23 28 0 


13 

11 29 51 15 

11 

29 36 

57 

0 

10 54.5 


1801 

9 9 40 16 

,6 

0 6 36 

23 28 1 


14 

11 29 36 55 

11 

29 21 

31 

0 

11 44.9 


1802 

9 9 25 56! 

5 

29 51 10 

23 28 0 


15 11 29 22 36 

11 

29 6 

6 

0 

12 35.2 


1803 

9 9 11 37 

5 

29 35 45 

23 28 0 

B 

16 

0 0 7 35 

11 

29 49 

49 

0 

13 25.6 


B 1804 

9 9 56 26 

6 

0 19 28 

23 27 58 


17 

11 29 53 6 

11 

29 34 

24 

0 

15 15.9 


1805 

9 9 42 6 

6 

0 4 2 

23 27 55 


18 

11 29 38 47 

11 

29 18 

59 

0 

15 6.3 


1806 

9 9 27 48 

5 

29 48 38 

23 27 51 


.19 

11 29 24 27 

11 

29 3 

33 

0 

15 56.6 


1807 

9 9 13 29 

5 

29 33 13 

23 27 48 

B 

20 

0 0 9 17 

11 

29 47 

17 

0 

16 47 


B 1808 

9 9 58 17 

6 

0 16 48 

23 27 44 

B 

40 

0 0 18 33 

11 

29 34 

33 

0 

33 34 


1809 

9 9 43 57 

6 

0 1 31 

23 27 42 

B 

60 

0 0 27 50 

11 

29 21 

50 

0 

50 21 


1810 

9 9 29 37 

5 

29 45 57 

23 27 40 

B 

80 

0 0 37 6 

11 

29 9 

6 


7 8 


1811 

9 9 15 17 

5 

29 30 32 

! 23 27 39 

B 

10C 

0 0 46 23 

- 11 

28 56 

23 

1 

23 55 


B 1812 

9 10 0 5 

> 6 

0 14 15 

• 23 27 39 

B 20C 

10 1 32 46 

11 

27 52 

46 

1 2 

47 50 


1813 

9 9 45 45 

i 5 

29 58 49 

l 23 27 40 

B 30C 

10 2 19 9 

1 11 

26 49 

c 

1 4 

11 45 


1814 

9 9 31 25 

» 5 

29 43 26 

1 23 27 41 

B 40C 

) 0 3 5 32 

! 11 

25 45 

35 

l 5 

35 40 


1815 

» 9 9 17 5 

*5 

29 27 58 

1 23 27 43 

B 50C 

) 0 3 51 5£ 

ill 

24 41 

51 

, 6 

59 35 


B 1816 

:9 10 1 55 

16 

0 11 41 

23 27 46 









1817 

'9 9 47 35 

15 

29 56 1£ 

1 23 27 48 









1818 

» 9 9 33 15 

55 

29 40 5( 

) 23 27 50 









1818 

19 9 18 55 

5 5 

29 25 24 

1 23 27 52 









B 182l 

) 9 10 3 41 

6 

0 9 r 

1 23 27 52 









1821 

9 9 49 25 

>5 

.29 55 45 

i 23 27 51 









1841 

9 9 58 39 

) 5 

29 42 59 

1 23 27 59 

































264 


APPENDIX TO THE ASTRONOMY. 


TABLE IV. The Sun’s Mean Longitude and Anomaly for Months and Days. 


d 

January. 

o 

February. 


March. 

3 

£ 

April. 

c*> 

Longitude. 

Anomaly. 

Cfi 

Longitude. 

Anomaly. 

'i 

1 ongitude. 

Anomaly. 


Longitude. 

Anomaly. 


s. 

o 

' 

it 

3. 

O 

' 

" 


s. 

0 

/ 

" 

s 

_ 

/ 

' 


s. 

O 

' 

• " 

s. 

o 

' 

" 


s. 

o 

' 

" 

s. 

O 

> 

// 

1 

0 

0 

59 

8 

0 

0 59 

8 

1 

1 

1 

32 

27 

1 

1 

32 

21 

i 

1 

29 

8 

20 

1 

29 

8 

9 

1 

2 

29 

41 

38 

2 

29 

41 

22 

2 

0 

1 

58 

17 

0 

1 

58 

17 

2 

1 

2 

31 

35 

1 

2 

31 

29 

2 

2 

0 

7 

28 

2 

0 

7 

17 

2 

3 

0 

40 

463 

0 

40 

30 

3 

0 

2 

57 

25 

0 

2 

57 

24 

3 

1 

3 

30 

43 

1 

3 

30 

37 

3 

2 

1 

6 

36 

2 

1 

6 

26 

3 

3 

1 

39 

553 

1 

39 

38 

4 

0 

3 

56 

33 

0 

3 

56 

32 

4 

1 

4 

29 

52 

1 

4 

29 

46 

4 

2 

2 

5 

45 

2 

2 

5 

34 

4 

3 

2 

39 

33 

2 

38 

46 

5 

0 

4 

55 

42 

0 

4 

55 

41 

5 

1 

5 

29 

0 

1 

5 

28 

53 

5 

2 

3 

4 

53 

2 

3 

4 

42 

5 

3 

3 

38 

113 

3 

37 

54 

6 

0 

5 

54 

50 

0 

5 

54 

49 

6 

1 

6 

28 

8 

1 

6 

28 

1 

6 

2 

4 

4 

1 

2 

4 

3 

49 

6 

3 

4 

37 

203 

4 

37 

3 

7 

0 

6 

53 

58 

0 

6 

53 

57 

7 

1 

7 

27 

17 

1 

7 

27 

10 

7 

2 

5 

3 

10 

2 

5 

2 

58 

7 

3 

5 

36 

28 3 

5 

36 

11 

8 

0 

7 

53 

7 

0 

7 

53 

6 

8 

1 

8 

26 

25 

1 

8 

26 

17 

8 

2 

6 

2 

18 

2 

6 

2 

6 

8 

3 

6 

35 

36'3 

6 

35 

18 

9 

0 

8 

52 

15 

0 

8 

52 

14 

9 

1 

9 

25 

33 

1 

9 

25 

26 

9 

2 

7 

1 

26 

2 

7 

1 

14 

9 

3 

7 

34 

4513 

7 

34 

27 

10 

0 

9 

51 

23 

0 

9 

51 

21 

10 

1 

10 

24 

42 

1 

10 

24 

35 

10 

2 

8 

0 

35 

2 

8 

0 

22 

10 

3 

8 

33 

53|3 

8 

33 

35 

11 

0 

10 

50 

32 

0 

10 

50 

30 

11 

1 

11> 

23 

50 

1 

11 

23 

42 

11 

2 

8 

59 

43 

2 

8 

59 

30 

11 

3 

9 

33 

13 

9 

32 

43 

12 

0 

11 

49 

40 

0 

11 

49 

38 

12 

1 

12 

22 

58 

1 

12 

22 

50 

12 

2 

9 

58 

51 

2 

9 

58 

38 

12 

3 

10 

32 

103 

10 

31 

52 

13 

0 

12 

48 

48 

0 

12 

48 

46 

13 

1 

13 

22 

7 

1 

13 

21 

59 

13 

2 

10 

58 

0 

2 

10 

57 

47 

13 

3 

11 

31 

183 

11 

30 

59 

14 

0 

13 

47 

57 

0 

13 

47 

54 

14 

1 

14 

21 

15 

1 

14 

21 

7 

14 

2 

11 

57 

8 

2 

11 

56 

55 

14 

3 

12 

30 

263 

12 

30 

7 

15 

0 

14 

47 

5 

0 

14 

47 

2 

15 

1 

15 

20 

23 

1 

15 

20 

15 

15 

2 

12 

56 

16 

2 

12 

56 

3 

15 

3 

13 

29 

3513 

13 

29 

16 

16 

0 

15 

46 

13 

0 

15 

46 

10 

16 

1 

16 

19 

31 

1 

16 

19 

23 

16 

2 

13 

55 

25 

2 

13 

55 

11 

16 

3 

14 

28 

433 

14 

28 

24 

17 

0 

16 

45 

22 

0 

16 

45 

19 

17 

1 

17 

18 

40 

1 

17 

18 

31 

17 

2 

14 

54 

33 

2 

14 

54 

19 

17 

3 

15 

27 

51 

3 

15 

27 

32 

18 

0 

17 

44 

30 

0 

17 

44 

27 

18 

1 

18 

17 

48 

1 

18 

17 

39 

18 

2 

15 

53 

41 

2 

15 

53 

27 

18 

3 

16 

27 

0 

3 

16 

26 

40 

19 

0 

18 

43 

38 

0 

18 

43 

35 

19 

1 

19 

16 

56 

1 

19 

16 

47 

19 

2 

16 

52 

50 

2 

16 

52 

36 

19 

3 

17 

26 

8 

3 

17 

25 

48 

20 

0 

19 

42 

47 

0 

19 

42 

43 

20 

1 

20 

16 

5 

1 

20 

15 

56 

20 

2 

17 

51 

58 

2 

17 

51 

44 

20 

3 

18 

25 

16 

3 

18 

24 

56 

21 

0 

20 

41 

55 

0 

20 

41 

51 

21 

1 

21 

15 

13 

1 

21 

15 

4 

21 

2 

18 

51 

6 

2 

18 

50 

51 

21 

3 

19 

24 

25 

3 

19 

24 

5 

22 

0 

21 

41 

3 

0 

21 

40 

59 

22 

1 

22 

14 

21 

1 

22 

14 

11 

22 

2 

19 

50 

15 

2 

19 

50 

0 

22 

3 

20 

23 

38 3 

20 

23 

13 

23 

0 

&2 

40 

12 

0 

22 

40 

8 

23 

1 

23 

13 

30 

1 

23 

13 

20 

23 

2 

20 

49 

23 

2 

20 

49 

8 

23 

3 

21 

22 

41 3 

21 

22 

21 

24 

0 

23 

39 

20 

0 

23 

39 

16 

24 

1 

24 

12 

38 

1 

24 

12 

28 

24 

2 

21 

48 

31 

2 21 

48 

16 

24 

3 22 

21 

503 

22 

21 

29 

25 

0 

24 

38 

28 

0 

24 

38 

23 

25 

1 

25 

11 

46 

1 

25 

11 

36 

25 

2 

22 

47 

40 

2 

22 

47 

25 

25 

3 

23 

20 

373 

1 

23 

20 

37 

26 

0 

25 

37 

37 

0 

25 

37 

32 

26 

1 

26 

10 

55 

1 

26 

10 

45 

26 

2 

23 

46 

48 

2 

23 

46 

33 

26 

3 

24 

20 

6 l 3 24 

19 

45 

27 

0 

26 

36 

45 

0 

26 

36 

40 

27 

1 

27 

10 

3 

1 

27 

9 

52 

27 

2 

24 

45 56 

2 

24 

45 

40 

27 

3 

25 

19 

153 

25 

18 

54 

28 

0 

27 

35 

53 

0 

27 

35 

48 

28 

1 

28 

9 

11 

1 

28 

9 

0 

28 

2 

25 

45 

5 

2 

25 

44 

53 

28 

3 

26 

18 

23 3 

26 

18 

2 

29 

0 

28 

35 

2 

0 

28 

34 

57 



In the months J anuary 

29 

2 

26 

44 

13 

2 

26 

43 

57 

29 

3 

27 

17 

31 3 

27 

17 

9 

30 

0 

29 

34 

10 

0 29 

34 

5 


and February of bissex- 

30 

2 

27 

43 

21 

2 

27 

43 

5 

30 

3 

28 

16 

40,3 

28 

16 

18 











tile years take away ( 

inp 





























Mir 



















31 

1 

p 

33 

18 

1 

0 

33 

12 


day from the time. 


31 

2 

28 

42 

30;2 

28 

42 

14 


















































SOLAR TABLES 


265 


The Sun's mean Longitude and Anomaly for Months and Days. 


o 

May. 

a 

June. 

O 

V 

July. 

G 

S- 

August. 

C/3 

Longitude. 

Anomaly. 

C fi 

Longitude. 

Anomaly. 

rn 

Longitude. 

Anomaly. 


Longitude. 

Anomaly. 


s. 

o 

/ 

'/ 

s. 

0 

/ 

n 


s. 

0 

/ 

// 

s. 

O 

/ 

// 


s. 

0 

/ 

II 

s. 

O 

1 

II 


s. 

0 

f 

II 

s 

O 

1 

If 

1 

3 

29 

15 

48 

3 

29 

15 

26 

i 

4 

29 

49 

6 

4 

29 

48 

39 

1 

5 

29 

23 

16 

5 

29 

22 

43 

1 

6 

29 

56 

34 

6 

29 

55 

55 

2 

4 

0 

14 

56 

4 

0 

14 

34 

2 

5 

0 

48 

15 

5 

0 

47 

47 

2 

6 

0 

22 

24 

6 

0 

21 

51 

2 

7 

0 

55 

43 

7 

0 

55 

4 

• 3 

4 

1 

14 

5 

4 

1 

13 

43 

3 

5 

1 

47 

23 

5 

1 

46 

55 

3 

6 

1 

21 

33 

6 

1 

21 

0 

3 

7 

1 

54 

51 

7 

1 

54 

12 

4 

4 

2 

13 

13 

4 

2 

12 

51 

4 

5 

2 

46 

31 

5 

2 

46 

3 

4 

6 

2 

20 

41 

6 

2 

20 

8 

4 

7 

2 

53 

59 

7 

2 

55 

20 

5 

4 

3 

12 

21 

4 

3 

11 

58 

5 

5 

3 

45 

40 

5 

3 

45 

12 

5 

6 

3 

19 

49 

6 

3 

19 

15 

5 

7 

3 

53 

8 

7 

3 

52 

29 

6 

4 

4 

11 

30 

4 

4 

11 

7 

6 

5 

4 

44 

48 

5 

4 

44 

20 

6 

6 

4 

18 

58 

6 

4 

18 

24 

6 

7 

4 

52 

16 

7 

4 

51 

37 

7 

4 

5 

10 

38 

4 

5 

10 

15 

7 

5 

5 

43 

56 

5 

5 

43 

28 

7 

6 

5 

18 

6 

6 

5 

17 

32 

7 

7 

5 

51 

24 

7 

5 

50 

44 

8 

4 

6 

9 

46 

4 

6 

9 

23 

8 

5 

6 

43 

5 

5 

6 

42 

37 

8 

6 

6 

17 

14 

6 

6 

16 

40 

8 

7 

6 

50 

33 

7 

6 

49 

54 

9 

4 

7 

8 

55 

4 

7 

8 

32 

9 

5 

7 

42 

13 

5 

7 

41 

44 

9 

6 

7 

16 

23 

6 

7 

15 

49 

9 

7 

7 

49 

41 

7 

7 

49 

1 

10 

4 

8 

8 

3 

4 

8 

7 

40 

10 

5 

8 

41 

21 

5 

8 

40 

52 

10 

6 

8 

15 

31 

6 

8 

14 

56 

10 

7 

8 

48 

49 

7 

8 

48 

9 

11 

4 

9 

7 

11 

4 

9 

6 

47 

11 

5 

9 

40 

30 

5 

9 

40 

1 

11 

6 

9 

14 

39 

6 

9 

14 

4 

11 

7 

9 

47 

58 

7 

9 

47 

18 

12 

4 

10 

6 

20 

4 

10 

5 

56 

12 

5 

10 

39 

38 

5 

10 

39 

8 

12 

6 

10 

13 

48 

6 

10 

13 

13 

12 

7 

10 

47 

6 

7 

10 

46 

25 

13 

4 

11 

5 

28 

4 

11 

5 

4 

13 

5 

11 

38 

46 

5 

11 

38 

16 

13 

6 

11 

12 

56 

6 

11 

12 

21 

13 

7 

11 

46 

14 

7 

11 

45 33 

14 

4 

12 

4 

36 

4 

12 

4 

12 

14 

5 

12 

37 

55 

5 

12 

37 

25 

14 

6 

12 

12 

4 

6 

12 

11 

29 

14 

7 

12 

45 

23 

7 

12 

44 

42 

15 

4 

13 

3 

45 

4 

13 

3 

21 

15 

5 

13 

37 

3 

5 

13 

36 

33 

15 

6 

13 

11 

13 

6 

13 

10 

38 

15 

7 

13 

44 

31 

7 

13 

43 

50 

16 

4 

14 

2 

53 

4 

14 

2 

28 

16 

5 

14 

36 

11 

5 

14 

35 

41 

16 

6 

14 

10 

21 

6 

14 

9 

45 

16 

7 

14 

43 

39 

7 

14 

42 

58 

17 

4 

15 

2 

1 

4 

15 

1 

36 

17 

5 

15 

35 

20 

5 

15 

34 

50 

17 

6 

15 

9 

29 

6 

15 

8 

53 

17 

7 

15 

42 

48 

7 

15 

42 

7 

18 

4 

16 

1 

10 

4 

16 

0 

45 

18 

5 

16 

34 

28 

5 

16 

33 

57 

18 

6 

16 

8 

38 

6 

16 

8 

2 

18 

7 

16 

41 

56 

7 

16 

41 

16 

19 

4 

17 

0 

18 

4 

16 

59 

53 

19 

5 

17 

33 

36 

5 

17 

33 

5 

19 

6 

17 

7 

46 

6 

17 

7 

10 

19 

7 

17 

41 

4 

7 

17 

40 

22 

20 

4 

17 

59 

26 

4 

17 

59 

1 

20 

5 

18 

32 

45 

5 

18 

32 

14 

20 

6 

18 

6 

54 

6 

18 

6 

18 

20 

7 

18 

40 

13 

7 

18 

39 

31 

21 

4 

18 

58 

35 

4 

18 

58 

10 

21 

5 

19 

31 

53 

5 

19 

31 

32 

21 

6 

19 

6 

3 

6 

19 

5 

26 

21 

7 

19 

39 

21 

7 

19 

38 

39 

22 

4 

19 

57 

43 

4 

19 

57 

17 

22 

5 

20 

31 

1 

5 

20 

30 

30 

22 

6 

20 

5 

11 

6 

20 

4 

34 

22 

7 

20 

38 

29 

7 

20 

37 

47 

23 

4 

20 

56 

51 

4 

20 

56 

25 

23 

5 

21 

30 

10 

5 

21 

29 

39 

23 

6 

21 

4 

19 

6 

21 

3 

42 

23 

7 

21 

37 

38 

7 

21 

36 

56 

24 

4 

21 

56 

0 

4 

21 

55 

34 

24 

5 

22 

29 

18 

5 

22 

28 

46 

24 

6 

22 

3 

28 

6 

22 

2 

51 

24 

7 

22 

36 

46 

7 

22 

36 

4 

25 

4 

22 

55 

8 

4 

22 

54 

42 

25 

5 

23 

28 

26 

5 

23 

27 

54 

25 

6 

23 

2 

36 

6 

23 

1 

59 

25 

7 

23 

35 

54 

7 

23 

35 

11 

26 

4 

23 

54 

16 

1 

23 

53 

50 

26 

5 

24 

27 

35 

5 

24 

27 

3 

26 

6 

24 

1 

44 

6 

24 

1 

7 

26 

7 

24 

35 

3 

7 

24 

34 

20 

27 

4 

24 

53 

25 

4 

24 

52 

58 

27 

5 

25 

26 

43 

5 

25 

26 

11 

27 

6 

25 

0 

53 

6 

25 

0 

15 

27 

7 

25 

34 

11 

7 

25 

33 

28 

28 

4 

25 

52 

33 

4 

25 

52 

6 

28 

5 

26 

25 

51 

5 

26 

25 

19 

28 

6 

26 

0 

1 

6 

25 

59 

231 

28 

7 

26 

33 

19 

7 

26 

32 

30 

29 

4 

26 

51 

41 

4 

26 

51 

14 

29 

5 

27 

24 

59 

5 

27 

24 

26 

29 

6 

26 

59 

9 

6 

26 

58 

31 

29 

7 

27 

32 

28 

7 

27 

31 

44 

30 

4 

27 

50 

50 

4 

27 

50 

23 

30 

5 

28 

24 

8 

5 

28 

23 

35 

30 

6 

27 

58 

18 

6 

27 

57 

40 | 

30 

7 

28 

31 

36 

7 

28 

30 

52 

31 

5 

CO 

ON* 

49 

58 

4 

CO 

49 

31 










31 

6 

28 

57 

26 

6 

to 

CO | 

56 

48j 

31 

7 

05 

CM 

30 

44 

7 

29 

30 

0 































































266 


APPENDIX TO THE ASTRONOMY. 


The Sun’s Mean Longitude and Anomaly for Months and Days. 


G 

September. 

o 

October. 


November. 

U> 

December. 

X 


Longitude 


Anomaly. 

X 

longitude. 

Anomaly. 


Longitude. 

Anomaly 

X 


Longitude 


Anomaly. 


S. 

o 

t 

// 

s. 

O 

/ 

// 


s. 

O 

/ 

// 

s. 

O 

/ 

ft 


s. 

0 

/ 

// 

s. 

0 

f 

// 


S. 

o 

/ 

// 

s. 

o 

/ 

// 

1 

8 

0 

29 

53 

8 

0 

29 

9 

1 

9 

0 

4 

3 

9 

0 

3 

13 

1 

10 

0 

37 

21 

10 

0 

36 

26 

1 

11 

0 

11 

31 

11 

0 

10 

30 

2 

8 

1 

29 

1 

8 

1 

28 

17 

2 

9 

1 

3 

11 

9 

1 

2 

21 

2 

10 

1 

36 

29 

10 

1 

35 

34 

2 

11 

1 

10 

39 

11 

1 

9 

38 

3 

8 

2 

28 

9 

8 

2 

27 

25 

3 

9 

2 

2 

19 

9 

2 

1 

2o 

3 

10 

2 

35 

38 

10 

2 

34 

43 

3 

11 

2 

9 

47 

11 

2 

8 

46 

4 

8 

3 

27 

18 

8 

3 

26 

33 

4 

9 

3 

1 

28 

9 

3 

0 

38 

4 

10 

3 

34 

46 

10 

3 

33 

50 

4 

11 

3 

8 

56 

11 

3 

7 

55 

5 

8 

4 

26 

26 

8 

4 

25 

41 

5 

9 

4 

0 

36 

9 

3 

59 

46 

5 

10 

4 

33 

54 

10 

4 

32 

58 


11 

4 

8 

4 

11 

4 

7 

3 

6 

8 

5 

25 

34 

8 

5 

24 

49 

6 

9 

4 

59 

44 

9 

4 

58 

54 

6 

10 

5 

33 

3 

10 

5 

32 

7 

6 

11 

5 

7 

12 

11 

5 

6 

11 

7 

8 

6 

24 

43 

8 

6 

23 

58 

7 

9 

5 

58 

53 

9 

5 

58 

2 

7 

10 

6 

32 

11 

10 

6 

31 

15 

7 

11 

6 

6 

21 

11 

6 

5 

20 

8 

8 

7 

23 

51 

8 

7 

23 

6 

8 

9 

6 

58 

1 

9 

6 

57 

10 

8 

10 

7 

31 

19 

10 

7 

30 

23 

8 

11 

7 

5 

29 

11 

7 

4 

27 

9 

8 

8 

22 

59 

8 

8 

22 

14 

9 

9 

7 

57 

9 

9 

7 

56 

18 

9 

10 

8 

30 

27 

10 

8 

29 

30 

9 

11 

8 

4 

37 

11 

8 

3 

35 

10 

8 

9 

22 

8 

8 

9 

21 

22 

10 

9 

8 

56 

18 

9 

8 

55 

27 

10 

10 

9 

29 

36 

10 

9 

28 

39 

10 

11 

9 

3 

46 

11 

9 

2 

44 

11 

8 

10 

21 

16 

8 

10 

20 

30 

11 

9 

9 

55 

26 

9 

9 

54 

35 

11 

10 

10 

28 

44 

10 

10 

27 

47 

1 i 

11 

10 

2 

54 

11 

10 

1 

52 

12 

8 

11 

20 

24 

8 

11 

19 

38| 

12 

9 

10 

54 

34 

9 

10 

53 

42 

12 

10 

11 

27 

52 

10 

11 

26 

55 

12 

11 

11 

2 

2 

11 

11 

1 

0 

13 

8 

12 

19 

33 

8 

12 

18 

47 

13 

9 

11 

53 

43 

9 

11 

52 

51 

13 

10 

12 

27 

1 

10 

12 

26 

4 

13 

11 

12 

1 

11 

11 

12 

0 

8 

14 

8 

13 

18 

41 

8 

13 

17 

55 

14 

9 

12 

52 

51 

9 

12 

51 

59 

14 

10 

13 

26 

9 

10 

13 

25 

12 

14 

11 

13 

0 

19 

11 

12 

59 

16 

15 

8 

14 

17 

49 

8 

14 

17 

2 

15 

9 

13 

51 

59 

9 

13 

51 

7 

15 

10 

14 

25 

17 

10 

14 

24 

19 

15 

11 

13 

59 

27 

11 

13 ,58 

24 

16 

8 

15 

16 

58 

8 

15 

16 

11, 

16 

9 

14 

51 

8 

9 

14 

50 

16 

16 

10 

15 

24 

26 

10 

15 

23 

28 

16 

11 

14 

58 

36 

11 

V 

14 

57 

33 

17 

8 

16 

16 

6 

8 

16 

15 

19 

17 

9 

15 

50 

16 

9 

15 

49 

24 

17 

10 

16 

23 

34 

10 

16 

22 

36 

17 

11 

15 

57 

44 

11 

15 

56 

41 

18 

8 

17 

15 

14 

8 

17 

14 

27 

18 

9 

16 

49 

24 

9 

16 

48 

31 

18 

10 

17 

22 

42 

10 

17 

21 

44 

18 

11 

16 

56 

52 

11 

16 

55 

48 

19 

8 

18 

14 

23 

8 

18 

13 

35 

19 

9 

17 

48 

33 

9 

17 

47 

40 

19 

10 

18 

21 

50 

10 

18 

20 

52 

19 

11 

17 

56 

1 

11 

17 

54 

57 

20 

8 

19 

13 

31 

8 

19 

12 

43 

20 

9 

18 

47 

41 

9 

18 

46 

48 

20 

10 

19 

20 

59 

10 

19 

20 

0 

20 

11 

18 

55 

9 

11 

18 

54 

5 

21 

8 

20 

12 

39 

8 

20 

11 

i 

51 

21 

9 

19 

46 

49 

9 

19 

45 

56 

21 

10 

20 

20 

7 

10 

20 

19 

8 

21 

11 

19 

54 

17 

11 

19 

53 

13 

22 

8 

21 

11 

48 

8 

21 

11 

0 

22 

9 

20 

45 

58 

9 

20 

45 

5 

22 

10 

21 

19 

16 

10 

21 

18 

17 

22 

11 

20 

53 

26 

11 

20 

52 

22 

23 

8 

22 

10 

56 

8 

22 

10 

8 

23 

9 

21 

45 

6 

9 

21 

44 

13 

23 

10 

22 

18 

24 

10 

22 

17 

25 

23 

11 

21 

52 

34 

11 

21 

51 

29 

24 

8 

23 

10 

4 

8 

23 

9 

16 

24 

9 

22 

44 

14 

9 

22 

43 

20 

24 

10 

23 

17 

32 

10 

23 

16 

33 

24 

11 

22 

51 

42 

11 

22 

50 

37 

25 

8 

24 

9 

13 

8 

24 

8 

25 

25 

9 

23 

43 

23 

9 

23 

42 

29 

25 

10 

24 

16 

41 

10 

24 

15 

41 

25 

11 

23 

50 

51 

11 

23 

49 

46 

26 

8 

25 

8 

21 

8 

25 

7 

32 

26 

9 

24 

42 

31 

9 

24 

41 

37 

26 

10 

25 

15 

49 

10 

25 

14 

49 

26 

11 

24 

49 

59 

11 

24 

48 

54 

27 

8 

26 

7 

29 

8 

26 

6 

40 

27 

9 

25 

41 

39 

9 

25 

40 

45 

27 

10 

26 

14 

57 

10 

26 

13 

57 

27 

11 

25 

49 

7 

11 

25 

48 

2 

28 

8 

27 

6 

38 

8 

27 

5 

49 

28 

9 

26 

40 

48 

9 

26 

39 

54 

28 

10 

27 

14 

6 

10 

27 

13 

6 

28 

11 

26 

48 

16 

11 

26 

47 

11 

29 

8 

28 

5 

46 

8 

28 

4 

57 

29 

9 

27 

39 

56 

9 

27 

39 

2 

29 

10 

28 

13 

14 

10 

28 

12 

14 

29 

11 

27 

47 

24 

11 

27 

46 

18 

30 

8 

29 

4 

54 

8 

29 

4 

5 

30 

9 

28 

39 

4 

9 

28 

38 

9 

30 

10 

29 

12 

22 

10 

29 

11 

22 

30 

11 

28 

46 

32|ll 

28 

45 

26 










31 

9 

29 

38 

13 

9 

29 

37 

18 










31 

11 

25 

45 

41 

11 

29 

44 

35 






































































SOLAR TABLES 


267 


TABLE V. Of the Sun's TABLE VI. Equations of the Sun's Centre. 

Mean Longitude and Jlnom- 

alyfor Hours , Minutes , and Argument. Sun’s Mean Anomaly. 

Seconds. 


Signs. — 

-0 


— 1 



— 2 



-3 



■4 


— 5 



0 

0 

/ 

0 

0 

/ 

'/ 

o 

/ 

// 

O 

/ ~ 

// 

0 

/ 

// 

O 

/ 

n 

O 

0 

0 

0 

0 

0 

56 

47 

1 

39 

6 

1 

55 

37 

1 

41 

12 

0 

58 

53 

30 

1 

0 

1 

59 

0 

58 

30 

1 

40 

7 

1 

55 

39 

1 

40 

12 

0 

57 

7 

29 

2 

0 

3 

57 

1 

0 

12 

1 

41 

6 

1 

55 

38 

1 

39 

10 

0 

55 

19 

28 

3 

0 

5 

56 

1 

1 

53 

1 

42 

3 

1 

55 

36 

1 

38 

6 

0 

53 

30 

27 

4 

0 

7 

54 

1 

3 

33 

1 

42 

59 

1 

55 

31 

1 

37 

0 

0 

51 

40 

26 

5 

0 

9 

52 

1 

5 

12 

1 

43 

52 

1 

55 

24 

1 

35 

52 

0 

49 

49 

25 

6 

0 

11 

50 

1 

6 

50 

1 

44 

44 

1 

55 

15 

1 

34 

43 

0 

47 

57 

24 

7 

0 

13 

48 

1 

8 

27 

1 

45 

34 

1 

55 

3 

1 

33 

32 

0 

46 

5 

23 

8 

0 

15 

46 

1 

10 

2 

1 

46 

22 

1 

54 

50 

1 

32 

19 

0 

44 

11 

22 

9 

0 

17 

43 

1 

11 

36 

1 

47 

8 

1 

54 

35 

1 

31 

4 

0 

42 

16 

21 

10 

0 

19 

40 

1 

13 

9 

1 

47 

53 

1 

54 

17 

1 

29 

47 

0 

40 

21 

20 

11 

0 

21 

37 

1 

14 

41 

1 

48 

35 

1 

53 

57 

1 

28 

29 

0 

38 

25 

19 

12 

0 

23 

33 

1 

16 

11 

1 

49 

15 

1 

53 

36 

1 

27 

9 

0 

36 

28 

18 

13 

0 

25 

29 

1 

17 

40 

1 

49 

54 

1 

53 

12 

1 

25 

48 

0 

34 

30 

17 

14 

0 

27 

25 

1 

19 

8 

1 

50 

30 

1 

52 

46 

1 

24 

25 

0 

32 

32 

16 

15 

0 

29 

20 

1 

20 

34 

1 

51 

5 

1 

52 

18 

1 

23 

0 

0 

30 

33 

15 

16 

0 

31 

15 

1 

21 

59 

1 

51 

37 

1 

51 

48 

1 

21 

34 

0 

28 

33 

14 

17 

0 

33 

9 

1 

23 

22 

1 

52 

8 

1 

51 

15 

1 

20 

6 

0 

26 

33 

13 

18 

0 

35 

2 

1 

24 

44 

1 

52 

36 

1 

50 

41 

1 

18 

36 

0 

24 

33 

12 

19 

0 

36 

55 

1 

26 

5 

1 

53 

3 

1 

50 

5 

1 

17 

5 

0 

22 

32 

11 

20 

0 

38 

47 

1 

27 

24 

1 

53 

27 

1 

49 

26 

1 

15 

33 

0 

20 

30 

10 

21 

0 

40 

39 

1 

28 

41 

1 

53 

50 

1 

48 

46 

1 

13 

59 

0 

18 

28 

9 

22 

0 

42 

30 

1 

29 

57 

1 

54 

10 

1 

48 

3 

1 

12 

24 

0 

16 

26 

8 

23 

0 

44 

20 

1 

31 

11 

1 

54 

28 

1 

47 

19 

1 

10 

47 

9 

14 

24 

7 

24 

0 

46 

9 

1 

32 

24 

1 

54 

44 

1 

46 

32 

1 

9 

9 

0 

12 

21 

6 

25 

0 

47 

57 

1 

33 

35 

1 

54 

58 1 

45 

44 

1 

7 

29 

0 

10 

18 

5 

26 

0 

49 

45 

1 

34 

45 

1 

55 

10|l 

44 

53 

1 

5 

49 

0 

8 

14 

4 

27 

0 

51 

32 

1 

35 

53 

1 

55 

2 o;i 

44 

1 

1 

4 

7 

0 

6 

11 

3 

28 

0 

53 

18 

1 

36 

59 

1 

55 

28|1 

43 

7 

1 

2 

24 

0 

4 

7 

2 

29 

0 

55 

3 

1 

38 

3 

1 

55 

341 

I 

42 

10 

1 

0 

39 

0 

2 

4 

1 

30 

0 

56 

47 

1 

39 

6 

1 

55 

37] 1 

41 

12 

0 

58 

53 j0 

0 

0 

0 

Signs 

+ 

11 


+ 

10 


+ ! 

9 

+ 8 


+ 7 


+ 6 


j 

! 


H 

/ 

// 

t!f 

H 

0 

/ 

// 

f 

n 

n 

//' 

iv 

// 

/ 

// 

/// 


iv 

V 

// 


iv 

i 

2 

27 

50 

31 

1 

16 

23 

2 

4 

55 

41 

32 

1 

18 

51 

3 

7 

23 

31 

33 

1 

21 

19 

4 

9 

51 

22 

34 

1 

23 

47 

5 

12 

19 

12 

35 

1 

26 

14 

6 

14 

47 

2 

36 

1 

28 

42 

7 

17 

14 

53 

37 

1 

31 

10 

8 

19 

42 

43 

38 

1 

33 

38 

9 

22 

10 

34 

39 

1 

36 

6 

10 

24 

38 

24 

40 

1 

38 

34 

11 

27 

6 

14 

41 

1 

41 

1 

12 

29 

34 

5 

42 

1 

43 

29 

13 

32 

1 

55 

43 

1 

45 

57 

14 

34 

29 

46 

44 

1 

48 

25 

15 

36 

57 

36 

45 

1 

50 

53 

16 

39 

25 

27 

46 

1 

53 

21 

17 

41 

53 

17 

47 

1 

55 

48 

18 

44 

21 

7 

48 

1 

58 

16 

19 

46 

48 

58 

49 

2 

0 

44 

20 

49 

16 

48 

50 

2 

3 

12 

21 

51 

44 

39 

51 

2 

5 

40 

22 

54 

12 

29 

52 

2 

8 

8 

23 

56 

40 

19 

53 

2 

10 

36 

24 

59 

8 

19 

54 

2 

13 

3 

25 

61 

36 

9 

55 

2 

15 

31 

26 

64 

3 

59 

56 

2 

17 

59 

27 

66 

31 

49 

57 

2 

20 

27 

28 

68 

59 

31 

58 

2 

22 

55 

29 

71 

27 

21 

59 

2 

25 

23 

30 

73 

55 

11 

60 

2 

27 

50 






















































268 


APPENDIX TO THE ASTRONOMY, 


TABLE VII. Logarithms of the Sun's Distance from the Earth . 


Argument. Sun’s Mean Anomaly. 


Signs 

0 

1 

2 

3 

4 

5 


0 

0 

5.007286 

5.006347 

5.603749 

5.000124 

4.996405 

4.993620 

O 

30 

1 

5.007285 

5.006284 

5.003640 

4.999995 

4.996292 

4.993555 

29 

2 

5.007282 

5.006220 

5.003531 

4.999867 

4.996180 

4.993491 

28 

3 

5.007277 

5.006154 

g.003420 

4.999738 

4.996069 

4.993429 

27 

4 

5.007269 

5.006087 

5.003307 

4.999611 

4.995959 

4.993369 

26 

5 

5.007260 

5.006018 

5.003194 

4.999483 

4.995850 

4.993311 

25 

6 

5.007249 

5.005946 

5.003080 

4.999354 

4.995742 

4.993255 

24 

7 

5.007235 

5.005872 

5.002965 

4.999227 

4.995636 

4.993201 

23 

8 

5.007218 

5.005797 

5.002849 

4.999099 

4.995531 

4.993150 

22 

9 

5.007200 

5.005720 

5.002732 

4.998971 

4.995427 

4.993102 

21 

10 

5.007180 

5.005642 

5.002614 

4.998844 

4.995325 

4.993055 

20 

11 

5.007158 

5.005562 

5.002495 

4.998717 

4.995224 

4.993009 

19 

12 

5.007134 

5.005480 

5.002375 

4.998590 

4.995126 

4.992966 

18 

13 

5.007107 

5.005397 

5.002254 

4.998463 

4.995028 

4.992926 

17 

14 

5.007079 

5.005312 

5.002134 

4.998336 

4.994932 

4.992888 

16 

15 

5.007048 

5.005225 

5.002012 

4-998210 

4.994836 

4.992852 

15 

16 

5.007015 

5.005136 

5.001890 

4.998084 

4.994743 

4.992818 

14 

17 

5.006980 

5.005047 

5.001767 

4.997960 

4.994652 

4.992786 

13 

18 

5.006943 

5.004956 

5.001643 

4.997837 

4.994562 

4.992757 

12 

19 

5.006905 

5 004863 

5.001518 

4.997714 

4.994474 

4.992731 

11 

20 

5.006864 

5.004768 

5.001393 

4.997591 

4.994387 

4.992706 

10 

21 

5.006821 

5.004672 

5.001268 

4.997468 

4.994302 

4.992683 

9 

22 

5.006776 

5.004575 

5.001142 

4.997347 

4.994219 

4.992663 

8 

23 

5.006730 

5.004477 

5.001016 

4.997226 

4.994138 

4.992646 

7 

24 

5.006681 

5.004377 

5.000889 

4.997106 

4.994058 

4.992631 

6 

25 

5.006630 

5.004275 

5.000762 

4.996987 

4.993980 

4.992618 

5 

26 

5.006577 

5.004173 

5.000635 

4.996868 

4.993904 

4.992607 

4 

27 

5.006522 

5.004069 

5.000508 

4.996750 

4.993831 

4.992599 

3 

28 

5.006466 

5.003963 

5.000380 

4.996634 

4.993759 

4.992593 

9_ 

29 

5.006408 

5.003857 

5.000252 

4.996519 

4.993688 

4.992590 

1 

30 

5.006347 

5.003749 

5.000124 

4.996405 

4.993620 

4.992589 

0 

Signs. 11 

10 

9 

8 

7 

6 

- 





























SOLAR TABLES 


269 



TABLE VIII. The Sun's Declination to every 
Degree of his Longitude. 


Argument. Sun’s Longitude. 



0 T 

North. 

1 a 

North. 

2 nN 

orth. 


Deg. 

6 £3= 

South. 

7 rri 

South. 

8 1 

South. 

Deg. 


O 

Declin. 

O 

Declin. 

O 

Declin. 


o 

/ 

/. 

0 

/ 

n 

0 

/ 

// 


0 

0 

0 

0 

11 

29 

5 

20 

10 

25 

30 

1 

0 

23 

53 

11 

50 

6 

20 

22 

57 

29 

2 

0 

47 

47 

12 

10 

56 

20 

35 

7 

28 

3 

1 

11 

39 

12 

31 

34 

20 

46 

55 

27 

4 

1 

35 

30 

12 

51 

59 

20 

58 

20 

26 

5 

1 

59 

20 

13 

12 

12 

21 

9 

21 

25 

6 

2 

23 

8 

13 

32 

12 

21 

19 

59 

24 

7 

2 

46 

54 

13 

51 

58 

21 

30 

13 

23 

8 

3 

10 

37 

14 

11 

30 

21 

40 

3 

22 

9 

3 

34 

17 

14 

30 

48 

21 

49 

29 

21 

10 

3 

57 

54 

14 

49 

52 

21 

58 

30 

20 

11 

4 

21 

27 

15 

8 

40 

22 

7 

6 

19 

12 

4 

44 

57 

15 

27 

13 

22 

15 

17 

18 

13 

5 

8 

22 

15 

45 

30 

22 

23 

3 

17 

14 

5 

31 

42 

16 

3 

31 

22 

30 

24 

16 

15 

5 

54 

57 

16 

21 

16 

22 

37 

18 

15 

16 

6 

18 

6 

16 

38 

44 

22 

43 

47 

14 

17 

6 

41 

9 

16 

55 

55 

22 

49 

50 

13 

18 

7 

4 

6 

17 

12 

48 

22 

55 

27 

12 

19 

7 

26 

57 

17 

29 

23 

23 

0 

38 

11 

20 

7 

49 

41 

17 

45 

40 

23 

5 

22 

10 

21 

8 

12 

17 

18 

1 

38 

23 

9 

39 

9 

22 

8 

34 

45 

18 

17 

18 

23 

13 

29 

8 

23 

8 

57 

5 

18 

32 

38 

23 

16 

53 

7 

24 

9 

19 

17 

18 

47 

38 

23 

19 

50 

6 

25 

9 

41 

19 

19 

2 

18 

23 

22 

20 

5 

26 

10 

3 

12 

19 

16 

37 

23 

24 

22 

4 

27 

10 

24 

56 

19 

30 

35 

23 

25 

57 

3 

28 

10 

46 

30 

19 

44 

13 

23 

27 

5 

2 

29 

11 

7 

53 

19 

57 

30 

23 

27 

46 

1 

30 

11 

29 

5 

20 

10 

25 

23 

28 

0 

0 

D. 

O 

Declin. 

O 

Declin. 

O Declin. 

D. 


11 X South. 

10 ££■ South. 

9 vj 

South. 



5 rn;North. 

4 SI North. 

8 55 North. 



35* 




s 

e 


"S 

<u 

Cc 

s 

u 

s> 

e 

J: 


s 

CO 




X 

HH 

W 

J 

CQ 

< 


C3 

s 

o 

c 

< 

C 


a 

3 

cn 


a 

g 

3 

to 

— 

< 


CO 


a* 





Q o o o o 

° GO © < '—i 

Is 

©4 CO CO GO 


t GO GO CO GO 

! 3 


I O 

X 

" OJ ©t ©4 ©* 

a 

_ r- cn on os 



*3 


i 

^ co co co co 

& 

1 "" 1 t-I 

s 

_ o ©< ©* 


' CO GO GO CO 

o 

as 

- ©* ©< ©4 ©< 


^ CO iO t- 

r—* t-H r—1 

to CD tD CO 


03 0)00 
©< ©< GO GO 




5 co m co -h 
^ CO CO CO CO 


in co r* oo 
nt n( &( ©i 


o* ©< ©< e* 


lO i> o 

iO iO 


GO 

m m co co 


„ T#< Tt< lO lO 
' ©* ©} ©4 ©< 


©* ©4 09 ©< 


03 T-H iO 
rf to iO iO 


in in >n in 


«. CO GO GO ^ 
' ©( N Gl Gl 


©4 ©4 ©< 


t— t- oo 03 

T}< Tjl 


in in m in 


co 


o o o o 

T-I ©9 GO 

























































270 


APPENDIX TO THE ASTRONOMY 


TABIyE X. Of the Equation of Time fitted to each Degree of the Ecliptic. Place of the Jlpogee 3s. 9°. 


Argument. Sun’s Longitude. 


u 

T 0 

8 

1 

n 2 

Z5 3 

SI 4 

5 

-n- 

6 

"1 

7 

t 

8 

V5 

9 

■*2 

10 

X 

21 

orq 

+— 




-4 

4 

4 

+ 







— 

+ 

+ 

+ 


ra 

s. 

m. s. 

m. s. 

m. s. 

m. s. 

m. s. 

tn. 

s. 

rn. 

s. 

m. 

s. 

m. 

S. 

m. 

s. 

m. 

s. 

0 

7 

36 

1 

9:3 

51 

1 

13 

5 

57 

2 

20 

7 

38 

15 

31 

13 

33 

1 

11 

11 

28 

14 

19 

i 

7 

17 

1 

233 

47 

1 

26 

5 

59 

2 

4 

7 

58 

15 

39 

13 

17 

0 42jl 1 

45 

14 

13 

2 

6 

58 

1 

36 

3 

41 

1 

40 

6 

0 

1 

48 

8 

19 

15 

46 

13 

0 

— 

12 12 

1 

14 

6 

36 

39 

1 

48 

3 

37 

1 

55 

6 

1 

1 

31 

8 

40 

15 

52 

12 

42 

+ 

17 

12 

17 

13 

59 

46 

20 

2 

0 

3 

32 

2 

7 

6 

1 

1 

14 

9 

1 

15 

57 

12 

23 

0 

46 

12 

32 

13 

51 

56 

1 

2 

11 

3 

26 

2 

20 

6 

0 

0 

56 

9 

21 

16 

2 

12 

4 

1 

16 

12 

46 

13 

43 

65 

42 

2 

223 

19 

2 33 5 

59 

0 

38 

9 

41 

16 

6 

11 

44 

1 

45 

12 

59 

13 

34 

7 5 

24 

2 

323 

12 

2 

45 

5 

57 

0 

20 

10 

1 

16 

9 

11 

23 

2 

14 

13 

12 

13 

24 

8 

5 

5 

2 

423 

4 

2 

58 5 

54 

4 i 

10 

20 

16 

11 

11 

1 

2 

43 

13 

24 

13 

14 

94 

47 

2 

5! 

2 

56 

3 

11 5 

51 


-18 

10 

39 

16 

13 

10 

39 

3 

11 

13 

35 

13 

3 

10 

4 

28 

3 

02 

47 

3 

23 5 

47 

0 

37 

10 

57 

16 

13 

10 

16 

3 

39 

13 

45 

12 

51 

11 

4 

9 

3 

82 

38 

3 

35:5 

42 

0 

57 

11 

15 

16 

13 

9 

53 

4 

7 

13 

54 

12 

39 

12 

3 

50 

3 

162 

29 

3 

465 

37 

1 

17 

11 

33 

16 

12 

9 29 

4 

35 

14 

2 

12 

27 

13 

3 

32 

3 

23.2 

19 

3 

58 

5 

31 

1 

38 

11 

51 

16 

10 

9 

5 

5 

2 

14 

9 

12 

14 

14 

3 

133 

302 

1 

8 

4 

9 

5 

24 

1 

58 

12 

8 

16 

7 

8 

40 

5 

29 

14 

16 

12 

0 

15 

2 

55 

3 

36' 

1 

57 

4 

19 

5 

17 

2 

19 

12 

25 

16 

4 

8 

14 

5 

56 

14 

22 

11 

46 

16 2 

37 

3 

41 

1 

46 

4 

29 

5 

9 

2 

40 

12 

41 

16 

0 

7 

48 

6 

22 

14 

27 

11 

31 

17 

2 

19 

3 

46 

1 

35 

4 

39 

5 

1 

3 

1 

12 

57 

15 

55 

7 

22 

6 

48 

14 

31 

11 

16 

18 2 

1 

3 

50 

1 

23 

4 

48 

4 

52 

3 

22 

13 

12 

15 

49 

6 

55 

7 

13 

14 

35 

11 

1 

19 

1 

43 

3 

53 

1 

11 

4 

57 

4 

43 

3 

44 

13 

27 

15 

42 

6 

28 

7 

37 

14 

38 

10 

46 

20 

1 

26 

3 

56 0 

59 

5 

5 

4 

33 

4 

5 

13 

42 

15 

35 

6 

0 

8 

1 

14 

40 

10 

30 

21 

1 

9 

3 

580 

46 

5 

13 

4 

22 

4 

26 

13 

56 

15 

26 

5 

32 

8 

24 

14 

41 

10 

14 

22 

0 

52 

4 

00 

34 

5 

20 

4 

11 

4 

47 

14 

9 

15 

17 

5 

4 

8 

47 

14 

42 

9 

58 

23 

0 

36 

4 

liO 

21 

5 

27 

3 

50 

5 

9 

14 

21 

15 

7 

4 

36 

9 

9 

14 

41 

9 

41 

24 

0 

20 

4 

1 


- 8 

5 

33 

3 

46 

5 

30 

14 

33 

14 

56 

4 

8 

9 

31 

14 

40 

9 

24 

25 

4 

4 

4 

1 

+ 5 

5 

39 

3 

33 

5 

32 

14 

44 

14 

44 

3 

39 

9 

53 

14 

39 

9 

6 

26 


-11 

4 

00 

19 

5 

44 

3 

19 

6 

13 

14 

53 

14 

31 

3 

10 

10 

14 

14 

37 

8 

48 

27 

0 

2613 

59 0 

31 

5 

48 

3 

4 

6 

35 

15 

5 

14 

17 

2 

41 

10 

34 

14 

34 

8 

30 

28 

0 

40 

3 

570 

46 

5 

52 

2 

50 

6 

56 

15 

14 

14 

3 

2 

11 

10 

53 

14 

30 

8 

12 

29 

0 

53 

3 

54!0 

59 

5 

55 

2 

35 

7 

17 

15 

23 

13 

48 

1 

41 

11 

11 

14 

25 

7 

54 

30 

1 

9 

3 

51 

1 

13 

5 

57 

2 

20 

7 

28l 15 

31 

13 

33 

1 

11 

11 

28 

14 

19 

7 

36 


The equations with -f are to be added to the apparent time to have the mean time; those with 
are to be subtracted from the apparent for the mean time. 





































SOLAR TABLES 


271 


TABLE XI. The Sun's Longitude for every Day in the Fear, at J\'uon. 


e 

p 

C/i 

January. 

February. 

March. 

April. 

May. 

June. 

July. 

August. 

Sept. 

October. 

Novcm. 

Decern. 


s. 

0 

t 

S. 

0 

f 

s. 

0 

f 

s. 

o 

/ 

s 

O 

/ 

s. 

O 

/ 

s. 

o 


s. 

O 

; 

s 

O 

/ 

3 

• 0 

/ 

s. 

O 

/ 

S 

O 

/ 

1 

9 

11 

21 

10 

12 

54 

11 

11 

8 

0 

11 

55 

1 

11 

12 

2 

11 

2 

3 

9 

41 

4 

9 

17 

5 

9 

7 

6 

8 

26 

7 

9 

15 

8 

9 

32 

2 

9 

12 

23 

10 

13 

5511 

12 

8 

0 

12 

54 

1 

12 

10 

2 

12 

0 

3 

10 

38 

4 

10 

14 

5 

10 

6 

6 

9 

25 

7 

10 

15 

8 

10 

33 

3 

9 

13 

24 

10 

14 

56lll 

13 

8 

0 

13 

53 

1 

13 

9 

2 

12 

57 

3 

11 

35 

4 

11 

11 

5 

11 

4 

6 

10 

24 

7 

11 

16 

8 

11 

34 

4 

9 

14 

25 

10 

15 

57 

11 

14 

8 

0 

14 

52 

1 

14 

7 

2 

13 

54 

3 

12 

32 

4 

12 

9 

5 

12 

2 

6 

11 

24 

7 

12 

16 

8 

12 

35 

5 

9 

15 

26 

10 

16 57 

11 

15 

8 

0 

15 

51 

1 

15 

5 

2 

14 

52 

3 

13 

30 

4 

13 

6 

5 

13 

0 

6 

12 

23 

7 

13 

16 

8 

13 

36 

6 

9 

16 

27 

10 

17 

58 

11 

16 

8 

0 

16 

50 

1 

16 

3 

2 

15 

49 

3 

14 

27 

4 

14 

4 

5 

13 

59 

6 

13 

22 

7 

14 

16 

8 

14 

37 

7 

9 

17 29 

10 

18 

59 

11 

17 

8 

0 

17 49 

1 

17 

1 

2 

16 

46 

3 

15 

24 

4 

15 

2 

5 

14 

57 

6 

14 

21 

7 

15 

17 

8 

15 

38 

8 

9 

18 

30 

10 

20 

0 

11 

18 

8 

0 

18 

48 

1 

17 59 

2 

17 44 

3 

16 

21 

4 

15 

59 

5 

15 

55 

6 

15 

21 

7 

16 

17 

8 

16 

39 

9 

9 

19 

31 

10 

21 

0 

11 

19 

8 

0 

19 

47 

1 

18 

56 

2 

18 

41 

3 

17 

18 

4 

16 57 

5 

16 

53 

6 

16 

20 

7 

17 

17 

8 

17 40 

10 

9 

20 

32 

10 

22 

1 

11 

20 

8 

0 

20 

45 

1 

19 

54 

2 

19 

39 

3 

18 

15 

4 

17 54 

5 

17 52 

6 

17 

19 

7 

18 

18 

8 

18 

41 

11 

9 

21 

33 

10 

23 

1 

11 

21 

7 

0 21 

44 

1 

20 

52 

2 

20 

35 

3 

19 

13 

4 

18 

52 

5 

18 

50 

6 

18 

19 

7 19 

18 

8 

19 

42 

12 

9 

22 

34 

10 

24 

2 

11 

22 

7 

0 

22 

43 

1 

21 

50 

2 

21 

33 

3 

20 

10 

4 

19 

49 

5 

19 

49 

6 

19 

i 8 

7 20 

19 

8 

20 

43 

13 

9 

23 

35 

10 

25 

3 

11 

23 

7 

0 

23 

41 

1 

22 

48 

2 

22 

30 

3 

21 

7 

4 

20 47 

5 

20 

47 

6 

20 

18 

7 

21 

19 

8 

21 

44 

14 

9 

24 

36 

10 

26 

3 

11 

24 

6 

0 

24 

40 

1 

23 

45 

2 

23 

28 

3 

22 

4 

4 

21 

45 

5 

21 

46 

6 

21 

17 

7 22 

20 

8 

22 

45 

15 

9 

25 

37 

10 27 

4 

11 

25 

6 

0 

25 

39 

1 

24 

43 

2 

24 

25 

3 

23 

2 

4 

22 

42 

5 

22 

44 

6 

22 

17 

7 23 

20 

8 

23 

47 

16 

9 

26 

39 

10 

28 

4 

11 

2<] 

6 

0 

26 

37 

1 

25 

41 

2 

25 

22 

3 

23 

59 

4 

23 

40 

5 

23 

43 

6 

23 

17 

7 24 

21 

8 

24 

48 

17 

9 

27 39 

10 

29 

4 

11 

27 

5 

0 

27 

36 

1 

26 

39 

2 

26 

19 

3 

24 

56 

4 

24 

38 

5 

24 

42 

6 

24 

16 

7 25 

21 

8 

25 

49 

18 

9 

28 

41 

11 

0 

5 

11 

28 

5 

0 

28 

34 

1 

27 36 

2 

27 

17 

3 

25 

53 

4 

25 

36 

5 

25 

40 

6 

25 

16 

7 

26 

22 

8 

26 

50 

19 

9 

29 

42 

11 

1 

5 

11 

29 

4 

0 

29 

33 

1 

28 

34 

2 

28 

14 

3 «6 

51 

4 

26 

33 

5 

26 

39 

6 

26 

16 

7 27 23 

8 

27 51 

20 

10 

0 

43 

11 

2 

6 

0 

0 

4 

1 

0 

31 

1 

29 

32 

2 

29 

11 

3 

27 48 

4 

27 31 

5 

27 38 

6 27 

16 

7 28 

23 

8 

28 

52 

21 

10 

1 

44 

11 

3 

6 

0 

1 

3 

1 

1 

30 

2 

0 29 

3 

0 

8 

3 

28 

45 

4 

28 

29 

5 

28 

37 

6 

28 

15 

7 29 

24 

8 

29 

54 

22 

10 

2 

45 

11 

4 

6 

0 

2 

3 

1 

2 

28 

2 

1 

27 

3 

1 

6 

3 

29 

43 

4 

29 

27 

5 

29 

35 

6 29 

15 

8 

0 

25 

9 

0 

55 

23 

10 

3 

46 

11 

5 

7 

0 

3 

2 

1 

3 

26 

2 

2 

25 

3 

2 

3 

4 

0 

40 

5 

0 

25 

6 

0 

34 

7 

0 

15 

8 

1 

26 

9 

1 

56 

24 

10 

4 47 

11 

6 

7 

0 

4 

1 

1 

4 

25 

2 

3 

22 

3 

3 

0 

4 

1 

37 

5 

1 

23 

6 

1 

33 

7 

1 

15 

8 

2 

26 

9 

2 

57 

25 

10 

5 

48 

11 

7 

7 

0 

5 

1 

1 

5 

23 

2 

4 

20 

3 

3 

57 

4 

2 

35 

5 

2 

21 

6 

2 

32 

7 

2 

1-5 

8 

3 

27 

9 

3 

58 

26 

10 

6 49 

11 

8 

7 

0 

6 

0 

l' 

6 

21 

2 

5 

17 

3 

4 

55 

4 

3 

32 

5 

3 

19 

6 

3 

31 

7 

3 

15 

8 

4 

28 

9 

4 

59 

27 

10 

7 50 

11 

9 

8 

0 

6 

59 

1 

7 20 

2 

6 

15 

3 

5 

52 

4 

4 

29 

5 

4 

17 

6 

4 

30 

7 

4 

15 

8 

5 

29 

9 

6 

1 

28 

10 

8 

51 

11 

10 

8 

0 

7 59 

1 

8 

18 

2 

7 12 

3 

6 

49 

4 

5 

27 

5 

5 

15 

6 

5 

29 

7 

5 

15 

8 

6 

30 

9 

7 

2 

29 

10 

9 

52 




0 

8 

58 

1 

9 

16 

2 

8 

10 

3 

7 46 

4 

6 

24 

5 

6 

13 

6 

6 28 

7 

6 

15 

8 

7 31 

9 

8 

3 

30 

10 

10 

52 




0 

9 

57 

1 

10 

14 

2 

9 

7 

3 

8 

43 

4 

7 22 

5 

7 

11 

6 

7 27 

7 

7 

15 

8 

8 

32 

) 

9 

4 

31 

10 

11 

53 




0 

10 

56 




2 

10 

5 


4 

8 

1 1 
<o I 

5 

8 

9 


r 

8 

15 


) 

10 

5 















































































APPENDIX TO THE ASTRONOMY. 




LUNAR TABLES- 


TABLE I. Of the Moon's Mean Motions in 
Julian Years. 


Years. 

j) ’s M. Long. 

3) ’s M. Ano. 


3 £1 Retro. 



S. 

O 

/ 

// 

S. 

O 

/ 

n 

S. 

Q 

/ 

" 


1 

4 

9 

23 

5 

2 

28 

43 

15 

0 

19 

19 

43 


2 

8 

18 

46 

11 

5 

27 

26 

30 

1 

8 

39 

26 


3 

0 

28 

9 

16 8 

26 

9 

45 

1 

27 

59 

9 

B 

4 

5 

20 

42 

570 

7 

56 

54 

2 

17 

22 

3 


5 

10 

0 

6 

2 

3 

6 

40 

9 

3 

6 

41 

46 


6 

2 

9 

29 

7 

6 

5 

23 

23 

3 

26 

1 

29 


7 

6 

18 

52 

13 

9 

4 

6 

39 

4 

15 

21 

12 

B 

8 

11 

11 

25 

53 

0 

15 

53 

47 

5 

4 

44 

6 


9 

3 

20 

48 

59 

3 

14 

37 

3 

5 

24 

3 

49 


10 

8 

0 

12 

4 

6 

13 

20 

17 

6 

13 

23 

32 


11 

0 

9 

35 

9 

9 

12 

3 

32 

7 

2 

43 

15 

B 

12 

5 

2 

8 

50 

0 

23 

50 

17 

7 

22 

6 

9 


13 

9 

11 

31 

55 

3 

22 

33 

56 

8 

11 

25 

52 


14 

1 

20 

55 

1 

6 

21 

17 

11 

9 

0 

45 

35 


15 

6 

0 

18 

6 

9 

20 

0 

26 

9 

20 

5 

18 

B 

16 

0 

22 

51 

46 

1 

1 

47 

34 

10 

9 

28 

12 


17 

3 

2 

14 

52 

4 

0 

30 

50 

10 

28 

47 

55 


18 

7 

11 

37 

57 

6 

29 

14 4 

11 

18 

7 

38 


19 

11 

21 

1 

2 

9 

27 

57 

19 

0 

7 

27 

21 

B 

20 

4 

13 

34 

43 

1 

9 

44 

30 

0 

26 

50 

15 

B 

40 

8 

27 

9 

26 

2 

19 

28 

56 

1 

23 

40 

30 

B 

60 

1 

10 

44 

9 

3 

29 

13 

24 

2 

20 

30 

45 

B 

80 

5 

24 

18 

52 

5 

8 

57 

52 

3 

17 

21 

0 

B 

100 

10 

7 

53 

35 

6 

18 

42 

20 

4 

14 

11 

15 

B 

200 

8 

15 

47 

lOjl 

7 

24 

40 

8 

28 

22 

13 

B 

300 

6 

23 

40 

45 

7 

26 

7 

0 

1 

12 

33 

45 

B 

400 

5 

1 

34 

20 

2 

14 

49 

20 

5 

26 

45 

0 

B 

500 

3 

9 

27 

55 

9 

3 

31 

40 

10 

10 

56 

15 


TABLE II. The Moon's Mean Longitude and 
Anomaly for Current Years. 


A. D. 

Mean Long. 

Mean 

Anom. 

Long, ft. 


S. 

0 

/ 

U 

S. 

O 

/ 

n 

S. 

0 

/ 


1761 

7 

i 

8 

8 

10 

12 

34 

50 

2 

7 

33 

33 

1781 

11 

14 

42 

54 

11 

22 

19 

18 

1 

10 

43 

18 

1791 

7 

14 

54 

59 

6 

5 

39 

35 

6 

27 

19 

46 

1792 

0 

7 

28 

40 

9 

17 

26 

44 

6 

7 

56 

52 

1793 

4 

16 

51 

45 

0 

16 

9 

59 

5 

18 

37 

9 

1794 

8 

26 

14 

51 

3 

14 

53 

14 

4 

29 

17 

26 

1795 

1 

5 

37 

57 

6 

13 

36 

29 

4 

9 

57 

43 

B 1796 

5 

28 

11 

37 

9 

25 

23 

38 

3 

20 

34 

49 

1797 

10 

7 

34 

43 

0 

24 

6 

53 

3 

1 

15 

6 

1798 

2 

16 

57 

48 

3 

22 

50 

8 

2 

11 

55 

23 

1799 

6 

26 

20 

54 

6 

21 

33 

23 

1 

22 

35 

40 

1800 

11 

5 

44 

0 

9 

20 

16 

38 

r 1 

3 

15 

57 

1801 

3 

15 

7 

5 

0 

18 59 

52 

0 

13 

56 

14 

1802 

7 

24 

30 

11 

3 

17 

43 

7 

11 

24 

36 

31 

1803 

0 

3 

53 

16 

6 

16 

26 

22 

11 

5 

16 

48 

B 1804 

4 

26 

26 

57 

9 

28 

13 

31 

10 

15 

53 

54 

1805 

9 

5 

50 

2 

0 

26 

56 

46 

9 

26 

34 

11 

1806 

1 

15 

13 

8 

3 

25 

40 

1 

9 

7 

14 

28 

1807 

5 

24 

36 

14 

6 

24 

23 

16 

8 

17 

54 

45 

B 1808 

10 

17 

9 

54 

10 

6 

10 

25 

7 

28 

31 

51 

1809 

2 

26 

33 

0 

1 

4 

53 

40 

7 

9 

12 

8 

1810 

7 

5 

56 

5 

4 

3 

36 

55 

6 

19 

52 

25 

1811 

11 

15 

19 

1 1 

7 

2 

20 

9 

6 

0 

32 

42 

B 1812 

4 

7 

52 

52 

10 

14 

7 

18 

5 

11 

9 

48 

1813 

8 

17 

15 

57 

1 

12 

50 

33 

4 

21 

50 

5 

1814 

0 

26 

39 

3 

4 

11 

33 

48 

4 

2 

30 

22 

1815 

5 

6 

2 

8 

7 

10 

17 

3 

3 

13 

10 

39 

B 1816 

9 

28 

35 

49 

10 

22 

4 

12 

2 

23 

47 

45 

1817 

2 

7 

58 

55 

1 

20 

47 

27 

2 

4 

28 

2 

1818 

6 

17 

22 

0 

4 

19 

30 

42 

1 

15 

8 

19 

1819 

10 

26 

45 

6 

7 

18 

13 

57 

0 

25 

48 

36 

R 1820 

3 

19 

18 

47 

11 

0 

1 

6 

0 

6 

25 

43 

1821 

7 

28 

41 

54 

1 

28 

44 

21 

11 

17 

5 

59 

1841 

0 

12 

16 

37 

3 

8 

28 

51 

10 

20 

15 

44 








































TABLE III. The Moon's Mean Motions for Months and Days. 


/ 


LUNAR TABLES. 269 




. co a o o ^ 

T- G9 CO CO 

iO »-0 CO U- t— 

CD CO 05 o O 

th Ol 09 CO 'O 1 

^ O iO tD O 

L— 


c; 

s C9 O th 09 

CO ^ O TH 

09 co o 

TH 09 CO O 

TH Cl CO ^ CO 

TH 09 CO ** 

iO 


• 

„ O CO h O CO 

CO 05 09 CD 05 

09 iO CO — 1-0 

CO h -t« L- h 

O CO CD 

O CO CO 05 09 

iO 


o 

^ hh rH 09 09 

09 G9 CO CO CO 

-t -r iO o 

lO TH 

TH TH 09 09 09 

CO CO CO GO .rf 

tH 



O CO GO GO CO GO 

CO CO CO CO CO 

CO ro CO CO co 

CO ^ tOH —fct 

Tf Tf Tf -H -f 

’i i -f 

Tf 



- h ^ >0) O CO 

h hi iC C5 CO 

l> th lO O CO 

o 

6 

9 

l 

L 

L- TH 05 09 

CD O -f CD 09 

CD 



tO *0 -T CO CO 

Cl Cl T-H 

id lO rr CO CO 

09 09 th 

iO O ^ CO CO 

09 09 th 

iO 



. co u- th 1.0 05 

CO ic* T-H LO 05 

09 CO O CO 

09 CO O ^ 00 

TH o 05 CO 1 - 

TT O 05 CO l- 

o 

• 

.w 

a 

uo o 

rr H Ol Cl Cl 

CO GO T^i *TJH T^ji 

O lO 

th th th 09 09 

60 CO GO ^ ^ 

lO 


3 

- GO CO O GO CD 

05 09 lO 0D T-H 

*t t* O CO CO 

05 09 CO 05 09 

O CD h L" 

O CO CD 05 09 

lO 

&H 

c 

-i ~ C9 

09 th 

T-H 09 TH Ol 

^ TH 09 TH 

09 09 th 

th 09 09 


a 

< 

• 







*H« 


^ CM C^ CO CO CO 

*^h rf lO iO O 

CO *o l> U- CO 

CO 05 05 05 O 

O TH TH O O 

TH TH rH 09 09' 

co 






TH 

TH — TH 




• 

- c^ ci r- o* 

r- Ol L- Ol t'- 

09 L- 09 09 

B- 09 L't 09 O- 

09 t- 09 l> 09 

U- 09 t- Ol fr 

CO 



^0 -* GI 

iO CO ^ ^ 

iO 09 CO t-h 

^ 09 O CO 

^ TH lO 09 

CO th Tj, 09 lO 

co 


o 

*. vO iC O O h 

CO 05 05 O 

O TH 09 09 CO 

co lO CO 

CD !> CO 05 

05 O O TH rH 

09 


-H 

CO ^ iO th 

Ol CO *0 1-H 

09 CO iO 

t-h 09 CO i-0 

i TH 09 CO ^ 

ID TH 09 CO 

IO 


o 

o O CO O O GO 

CO 05 09 O 05 

09 iO CO t-h iO 

00 TH ^ ^ O 

tBOCOO 

05 CO CD 05 09 

>o 


s 

— 09 09 

t— i 09 »—• 09 

09 TH T-H 

09 th 09 09 

rH tH 09 

09 TH 



—H 

C/2 09 09 CO CO ^ 

rf rj< lO iO CO 

CO t- CO CO 

CD 05 05 O O 

T- TH O O O 

TH th 09 09 CO 

CO 


*5 




TH rH 

—H TH 




th a* eo ^ io 

CO L- CO 05 O 

TH 09 CO lO 

CD lO* CO 05 O 

TH 09 CO ^ *-0 

CD h CO 05 O 

T-H 




T^ 

T^ T-* T-H »-H t-H 

t-H rH t-h rH 09 

09 09 09 09 09 

09 09 09 Ol CO 

CO 



. 

. O th 09 09 CO 

rf rf* lO iO CO 

i> !> CO 05 C5 

O TH H 09 CO 

CO ^ LO CD 

CD CD 



rO 

^-O 

" ^ th 09 

CO ^ ^C, ■ T-H 

09 CO ^ iO 

09 GO ^ iO 

th 09 CO iO 

th 09 



• 

*—« -rf CO — -rj* 

r- O GO 1> O 

CO CO 05 09 CO 

05 09 iO CO 09 

lO CO TH Tf t— 

th rf L't 



o 

r« rj« rr iO i£> 

O T-h 

T-H T-H T-H 09 09 

oi co co co rr 

rt lO LO lO 




S- 

O ^H TH 1 — TH T-H 

-H 09 09 09 09 

09 09 09 09 09 

09 09 0> 09 09 

09 09 09 G9 09 

CO CO CO 




^ i> rH LO O 09 

o O i 1 CO Cl 

CO O ^ CD 09 

CD O ^ CO Ol 

CD O ^ CO 09 

CD O ^ 



• 

" -o CO Cl 09 

T-H T-H iO i-O 

^ -t CO 09 09 

TH H lO lO 

^ CO 09 09 

TH TH 




^ ^ CO Cl o o 

•T CO 09 ^O 05 

CO 9> h iO 05 

CO h TH r^ 00 

09 D O -t CO 

09 CD O 


£■ 

X 

T-H T-H G9 

09 09 CO CO CO 

Tt Tf lfi iQ LO 

TH TH tH 

09 09 CO CO CO 

^ rf lO 


Ct 

— 

o 

CDih^L-O 

CO CO 05 09 iO 

CD ^ Tt j> O 

•t h O CO CD 

05 09 iO CD h 

^ O 


O 

a 

O Ol T-H 09 09 

t-h 09 t-h 09 

09 TH 

th 09 th 09 

TH rH 09 TH 

09 09 


Eh 

O 

< 

C/2 t- 09 09 CO CO 

^ ifl O 

CO CO IT- C"- CO 

OD CO 05 05 O 

O TH TH tH O 

D 1 TH tH 


r-r. 





TH 

rH TH TH TH 



h-* 

• 

- T-H CO TH CO »H 

CO T—H CO T-H CO 

H CO TH CO TH 

CD th CD th CD 

TH CD TH C- 09 

1> Ol L" 




" t-h lO 09 

GO hth rf 09 lO 

CO rH lO 

09 CO th th 

09 iO CO 

TH LO 09 




CO 05 Ci O ^ 

t-h 09 09 CO CO 

iO o CO CO 

t'- CO CO 05 05 

O O TH 09 09 

co co ^ 



o 

^ CO O vC ^ Cl 

CO ^ t-h 

09 GO iO 

TH 09 GO ^ lO 

th 09 CO ^ tO 

TH 09 



o 

o t-h rf t— i —1« 

O CO L* O 

CO CO 05 09 CO 

05 09 i-O CO th 

iD CO H t- 

H H [r 



s 

th 09 th 09 

09 t—h 

T-H 09 09 

TH TH 09 TH 

09 09 th 

TH 09 




m 09 09 09 GO GO 

rf -t iO iO CD 

co co r- r- od 

CO 05 05 05 O 

D5 H rH (D 1 

TH tH rH 







TH 

TH TH TH 



rvi 


i-i 09 CO tj« *0> 

CO CO 05 O 

h 09 CO iO 

CD C- CO 05 O 

H 09 CO ^ lO 

CD L* CO 




T-H 

T-H T-H rH hH rH 

TH tH tH th 0'9 

09 09 09 09 09 

09 09 09 




c? 

- ^ -H Cl CO GO 

^ ^ lO CO CO 

L* CO CO 05 O 1 

O th h 09 CO 

CO Trf lO lO CD 

L" CO CO 05 

o 


" th 09 CO ^ lO 

T-H 09 CO ^ 

iO th 09 TT 

lO th 09 CO 

^ lQ th 09 

CO ^ lO th 

CO 


• 

+-> 

^ CO CO O Cl LO 

05 09 i-O CO t-h 

^ CO TH ^ 

O rt< l> O CO 

CD 05 CO CD 05 

09 iO CO 09 iO 

CO 


o 

t-H rH 

t-h 09 09 09 CO 

CO CO rf TH 

iO lO iO 

rH rH th 

09 09 09 CO CO 

CO 



c/d o c o o o 

O O O O O 

o o o o o 

o o a th th 

rH rH H rH rH I 

H rH rH rH rH 

-H 



^ rt 00 09 CD O 

T-f CO 09 CO O 

CO h TH LO 05 

CO It- th lO 05 

GO B TH lO 05 

CO Jr- th iO 05 

CO 



O ’t ^ CO CO 

09 T-H T-H 

lO Hf CO 09 

09 th th lO 

LO ^ ^ CO 09 

09 TH TH lQ 

iO 

• 


GO h ^ »C Q 

CO l> TH lO 05 

09 CO O CO 

09 CD O -* t- 

TH lO 05 GO t'* 

^ lO 05 CO CD 

o 


<H 

H 

— H H rH 

09 09 CO GO CO 

rf ^ uO iD iO 

TH HH HH 

^ 09 09 CO CO 

^ ^ ^ iO iO 


rt 

H 

o 

n CO CO C5 09 iO 

CO HH t t- O 

CO CO 05 09 lO 

05 09 iO CO th 

rt 1 O GO CD 

05 09 uO CO rn 

IO 

S3 

a 

° ’HOI 09 

T-H T-H 09 T-H 

09 T- TH 

09 th 09 / 09 

TH TH 09 

09 rH 

TH 

C 

ct 

< 

(72 O O t— *h 09 

09 CO CO CO -f 

rf lO iO CO CO 

CD U- t- CO CO 

05 05 O O O 

^ ^ O O TH 








rH TH rH 

HH T-H 



• 

- iO O iO O R5 

OlOO VO O 

iO o O O i0 

O iO o »o th 

CD TH CD TH CD 

TH CD ' CD TH 

CD 


F* 

CO th rHi 09 lO 

CO T-H T-H LO 

09 CO th ^ 

09 O CO 

rH xD Ol CO 

TH 09 iO CO 



r« 

O th rH 09 09 

CO " 3 * rf O >0 

CO CO CO 

05 05 O H t 

09 09 CO -f 

iO iO CD CD 1> 

CD 


o 

H 

t—< 09 CO ^ *f5 

th 09 GO ^ 

-0 th 09 CO 

rf iO th 09 GO 

tT lO th 09 

CO ^ LO th 

09 


o 

0 GO CO 05 09 i0 

05 09 )0 CO t-h 

Tf CO th *rf IT- 

O CO B O CO 

CD 05 CO CD 05 

09 iD CO 09 iO 

CO 


s 

° ^ 09 09 

t-h TH 09 tH 

09 09 th 

TH 09 TH 09 

TH TH 09 

t-h 09 09 




m o o th —< 09 

09 CO CO CO ^ 

^ 1-0 lO CO co 

r- t- t- co co 

05 05 O O O 

^h th o O TH 

rH 







rH rH rH 

H 


rio 


t-i 09 CO ^ 

CO CO 05 O 

TH 09 CO ’■f lO 

CD B CO 05 O 

t 09 CO ^ iO 

CD r- CO 05 o 

TH 




TH th TH tH rH 

h,hh ^ c 9 

09 09 09 09 09 

09 09 09 09 GO 

CO 


3G| 


• < 


In January and tebruary ol a bissextile year, subtract 1 from the number of days 



















































































































TABLE III. The Moon's Mean Motions for Months and Day. 


270 


APPENDIX TO THE ASTRONOMY 



C§ 

„ t' CO CO 05 O 

O H H CO 

co -t lO lO CD 

P P CO CO 05 

O O -H G* G* 

CO rr 4 rf iO CD 



^ iO n-i g* rf 

iD Hi Dl CO 

tJ* lG G* 

CO ^ UO TH 

CO rf iO th 

G* CO rf iO 



• 

- G* D> CO OJ lO 

CO G} lO CO h 

p h ^ p 

O CO CD O CO 

CD 05 G* CD G 

G* UO CO hh >0 



o 

T-H HH 

TH G} G} G} co 

CO CO ^ ^ 

lG uD iG 

TH T— 1 rH 

G* G* G* CO CO 



£ 

0 CO CO CO CO CO 

CO CO CO CO CO 

CO CO CO CO CO 

CO CO CO 05 05 

05 05 05 05 05 

05 05 05 05 05 




. Oi O O ^ h 

H ID C5 CO h 

HH ID CO 00 P- 

rH lD G CO P 

hi iC G CO P- 

th lO 05 G* CD 



• 

" Tt CO CO O'} th 

T-h vO lO *t 

rf GO G* G* th 

t-h iD O ^ 

rf CO G^ G* th 

hh iO D ^ 




Cl CD C CO 

G} CD 05 GO P* 

hi iD G GO P 

HH ID CO G* CD 

O rf CO G* CD 

O -t P hh lO 



r2 

£ 

o 

" iD iO 

hH h-i r-H G-} 

co co co rf rf 

lO O iG 

HH TH TH G* G* 

CO CO CO rf rf 


QJ 

o >D CO G* *D CO 

h »t h o CO 

CD 05 G* lD CO 

rH *t P Hi -t 

p O CO CD 05 

G* iO CD h -ji 


C 

*—« 

hh th G* 

T-H 04 0 s * 

T-i G* HH G* 

G* HH TH 

G* th G* th 

TH G* r-i G* 


D 

>-“5 

c 

C/0 0 CD P* i> P“ 

CO 00 05 05 O 

O O ~ T-i o 

O TH th G* G* 

G* CO CO rf rf 

iD ^0 iO CD CD 





hh 

T-H T-H T-H T-H 






• 

^ rf 05 rf C5 rf 

05 ^ 05 iO O 

lD 0^0 0^0 

OiOO^OO 

iO O O iO 

O iO O xD o 




h G} 

CO T-H rf G* 

CO TH rr G< ID 

CO h iD 

0>* GO hh rf 

G* iO CO rf 



c 

_ CO 05 05 O ^ 

HH G* Gl CO -t 

rf lD iD CD CD 

P CO CO 05 G 

O th rH CN G* 

CO CO rf lO iD 




O ©1 CO 

T^ lO HH G* 

GO rf iD hh 

G* CO rf O 

G* CO rf iO 

hh G* CO rf iO 



O 

n C} uO 05 O} 

CO H o CO H 

P O ^ P 

O CO CD 05 CO 

CD 05 G* iO 05 

G* iO CO th rf 




° Cl T-i th 

C}hO} g* 

rH Hi O'* 

HH G* T-i 

HH G* HH G* 

G* HH th 



HH 

c/5 cd P- i> co co 

CO G 05 O O 

HH HH O O O 

H H O} c* CO 

CO CO rf rf iO 

lQ CD CD P P 





rH hH 

T-H T-H 





Days. 

t-h G* CO rf *D 

CD h OO G O 

H G* CO 't vO 

CD P CO G O 

hh G* CO rf iD 

CD P CO 05 O 


HH 

rH T-H T-H rH T—i 

Hi rH tH t-H G* 

G* G* G* G* G* 

G* G! G* G* CO 




• 

- D CO CO Q O 

O T-H G* G* CO 

^ Tt lD CD CO 

P P CO 05 05 

O HH rH G* CO 

CO rf rf iO CD 

CD 


c? 

G} CO ^ O ih 

G* CO rf iD 

HH O' * CO rf iD 

H G* CO 

TH G* CO rf 

O th G f CO 

rf 



. ^hOCOh 

O CO CD 05 CO 

CD 05 O* iO 00 

G* vD CO t-h -*i 

co TH rf P o 

CO P O CO CD 

05 


O 

Gl G* CO CO CO 

rf rf rf rf iD 

vD vD 

HH TH T—i G* G* 

G* CO CO CO rf 

rf ^ iO O iQ 

iD 


a 

O CO CO CO CO CO 

CO CO CD CD CO 

CD CD P P P 

P- P“ P- P- 

P P P P P 

p- P* 

p* 



1 . 05 CO C- rH lO 

05 CO L" h iD 

05 G* CD O ^ 

CO G* CD O ”t 

CO G* CD O rf 

CO G* CD O rf 

CO 


. 

' ^ ^ CO CO O} 

HH T-H lD 

^ ^ CO CO G* 

TH TH ID 

rf rf CO CO G* 

TH TH iO 

rf 



^ hh iD. 05 CO 1> 

th lD 05 CO CO 

O ^ CO G* CD 

O CO G* ID 

G CO P ^ iO 

O CO P h rf 

CO 


03 

C2 

lO lQ 

H H H Oi C} 

GO GO CO rf rf 

iD tD lD 

▼H tH G* G* 

G* CO CO rf rf 

rf 

b 

c 

o 

O CO CO o CO 

CO 05 G* iD CO 

H Tf i- O CO 

CD 05 G* CD 05 

Gl iO CD h if 

P O CO CD 05 

G* 


a 

0 *- T- 

GJ G* hh 

HH G* HH G* 

TH T-i G* 

HH G* G* 

Hi Hi G* 

G* 

a 

< 

C/3 rf iD iD CO CO 

CD l> Ch CO CO 

C5 o o o o 

tH rH (D O’ ‘O 

TH TH G* G* CO 

CO ^ rf rf iD 

>D 





T-H r-H 

TH th 




• 

- CO CO CO CO 05 

D Tt Q Tt 

G ^i G} ^ G 

rt G ^ G -t 

G rf G *t G 

rf 05 rf 05 rf 

C5 


PS 

CO hh rf G* *D 

CO ^ h lO 

G* CO hh rf 

G* iQ CO Tf 

HH IQ G* CO 

hh rf G* iD CO 



a 

o 

O H TH ^ Ol 

CO >h uG lG 

CD P P CD CO 

05 05 O th — h 

G* G* CO rf rf 

iD iD CD CD *> 

CO 


• H 
+- 

G* CO rf iD 

hh G* CO iO 

th G* CO rf 

lO G* CO rt 

iO th G* CO 

rf iO th g* 

CO 


O 

O rf p- O GO P- 

O CO CO 05 G* 

CD 05 G* iD CO 

rH lQ CO TH Hf 

P Hi Tf p O 

CO CD O CO CD 

05 



TH HH G* 

HH G* T-h 

T-H 0^* HH G* 

G^ Hi TH 

G* TH G* G* 

Hi TH G* 


a 

c/5 io o co co co 

P* CO CO 05 

G G O O ^ 

TH £0 O* TH TH 
tH 

TH G^ GA CO CO 

rf "f iO vO 

CD 

1 Da} r s. 

I *— 1 O'! CO ^ ilO 

CD P- CO C5 O 

TH G* CO rf lD 

CD P CO G O 

hh G* CO rf iO 

CD P CO 05 O 

T—i 


T-H 

rH tH tH th tH 

TH rH H TH G* 

G* G* G* G* G* 

G* G* G* G* CO 

CO 



• 

> CO 05 05 O rH 

Hi G* CO CO rf 

Tt ID CD CD P 

CO 00 05 O O 

HH T-H GJ CO CO 

rf if) lO CD P 



Co 


Hi G* CO rf 

ID TH G* CO 

rf lQ G* CO 

rf iO hh G* 

CO rf iD th 



• 

05 O) lO CO ^ 

i*G CO h -t P 

O P O CO 

CD 05 CO CD 05 

G* iO 05 GJ iO 

OO hh rf CO th 



o 

^ iO O lO 

t-H t-H rH 

G* GJ G* CO CO 

CO CO rf rf rf 

D iO D 

HH T-H T—i G^ 



rrb 

O ^ ^ ^ iT5 

lG iC vO lO 

iD ^D iD 

iO iO o 

iO iO iO CD CD 

CD CD CD CD CD 




^ O rf CO G* CD 

O ^ CO G* CD 

O ^ CO G* CD 

O rr P th lO 

G CO P hh iO 

05 CO P th iO 



• 

s 

^ iO rf CO CO G* 

G* HH ID 

iD ^ CO CO G* 

G* TH lO 

rf rf CO CO G* 

HH TH lD 



^ ^ CO 0 } CD O 

^ CO G* CD 05 

GO P hh iD 05 

CO P hi o CO 

G* CD O rf CO 

C} CD O rf p 


i • 

lO T-H 

HH T-H G* G* G* 

CO CO ^ ^ 

m io 

th th G* G* G* 

CO CO rf rf rf 


l"*? 

o 

_ CO t-h lD CO hh 

^ P* O CO CD 

05 G* iD CO hh 

rt 1 P h rt p 

O CO CD G G* 

iD CO HH rf p 


*H 

‘ CL, 
\< 

fl 

< 

0 T-H T-H OJ TH 

t/5 GO rf rf rf iD 

G* G^ HH 

CD CD P l> 

G* hh G* G* 

P CO CO 05 05 

HH H O'* 

O O T^ TH TH 

TH G* HH 

O O TH TH G* 

HH G* HH G* 

G* G* CO CO rf 







^ HH HH TH 





• 

. CO CO CO CO CO 

CO CO CO CO CO 

CO CO CO CO co 

CO CO CO CO CO 

CO CO co co co 

CO CO CO CO CO 


\ 


rf T-H lO G* 

CO TH T^ 

iD CO TfH 

ID G* CO hh 

rf G* iO CO 

^ th iD G* 



c 

- CO CO Tf Tf lO 

CD CD P P CO 

CO 05 o O hh 

th G* CO CO rf 

nf iD vO CD P 

P CO CO 05 O 



o 
*—« 

t-h G* CO rf 

ID H, G} CO 

^ lD h G* CO 

rf iO hi G* 

CO rf iO th 

G* CO rf iD hh 



o 

. 05 O} VO CO Hi 

^ OD h Tf p 

O CO P O CO 

CD 05 CO CD 05 

G* iC CO G* iO 

CO hh rf p th 



a 

° g* *h g* e* 

HH TH G* 

TH G* G* 

T-H G* TH O* 

G* TH T—i 

G* th G* G* 



a 

c/5 co rf rf id *d 

CD CD P P P 

CO CO 05 05 O 

O O hh th O 

O TH ^i G* G* 

CO CO rf rf 





HH 

HH rH t—i T-h 




Days. | 

Hi d CO ^ »o 

CD P CO 05 O 

HH GJ CO ^ 1-0 

CD P CO 05 o 

HH G* CO rf i-O 

CD P- CO 05 O 



L - ^ 

T" 1 ^ ^ *"* 

HH TH T-i — G* 

G* G* Gl G* G* 

G* G* G* G* CO 







































































































TABLE III. The Moon's Mean Motions for Months and Days. 


LUNAR TABLES 


•r 




27 i 


] 

• 

n CD CD l- CO CO 

CO O O t-h 04 

04 GO GO ^ lO 

k-O CD 15- CO 


04 04 GO rh 



cS 

- th (M CO ^ iO 

04 GO iO 

t-h 04 CO 

>lO T-h 04 go 

Tt kO T-H 04 GO 

^ kO th 04 




'O CO "t D* 

t-h rf r* O GO 

t- O GO CO 05 

04 CO C5 04 iO 

CO TH kO CO T-H 

rf l> th rt 1 !> 



o 

vO kO 

T-H T-h T-h 04 04 

04 GO GO GO GO 

rf rf Tf iO 

kQ rH 

▼H th 04 04 04 




0 04 04 CO CO GO 

GO GO GO GO GO 

CO GO GO GO GO 

GO GO GO GO GO 

GO T-f Tf -t< T-f 

rf rf rjr 




TH T—1 TH r—( r—f 

T-H l-H T-H T—1 T-H 

”—■ 4 ^H —' T-H T—( 

T-H T-—' t-H t-h t-H 

t-H T-H T-H T-H T-H 

rH rH H th rH 




^ OO^COO^ 

CO O *+ OD H 

CO D GO h H 

C5 GO h h 

kO 05 GO 1> T-H 

kO 05 GO TH 




04 04 th 

iO O ^ GO GO 

04 T-H T-H 

iO rf GO GO 

04 TH T-H 

kO rf rf GO GO 


fn 

f 0) 

Lo 

a 

Q 

rH lO O CO ^ 

O ""t* CO 04 CO 

O T-f CO 04 CO 

05 GO L- t-h iO 

05 GO — kO 

CO 04 CO O r* 


vO iO kO 

T-H t-h t-h 04 04 

• 

CO GO GO Tt ^ 

rf iO iD 

th rH 04 04 

04 CO CO ^ 


! a 

o 

O GO 1> O 

CO CO 05 04 kO 

CO H ^ h O 

GO CO 05 GO CO 

05 04 kO CO th 

Tt 1> O CO CO 


I 

fl 

<5 

° 04 th 

t-h 04 04 

T-H TH 04 TH 

04 rH t-h 

04 h 04 04 

TH rH 04 


o- 

I ^ 

rn O O th th O 

O O t-h t-h 04 

04 GO GO GO 

kO kO CO CO 

CO t- CO 00 

05 05 O O O 



v r—l rH r—( r—i 





H TH tH 


• 

0 t- C^ D- 04 t- 

04 t— 04 tG* 04 

4> 04 04 I> 

04 C- 04 C- 04 

1> GO CO GO CO 

CO CO GO CO GO 




04 GO th ^ 

04 uO CO ^ 

th lO 04 CO 

TH rf 04 kT5 GO 

^THk0 04 

GO th rt 04 



a 

04 CO GO 

iC lQ CO G- 

00 CO 05 O O 

th t-h 04 04 GO 

rf 1 rf kO kO CO 

L- t' CO CO 05 



o 

^ t-h 04 GO ^ 

4 O th 04 GO 

iO 04 GO 

T-f kQ T-H 04 

GO ^ kO rH 

04 GO kO 



O 

iO co h ^ t- 

O rt« i> O GO 

CO 05 GO CO 05 

04 kO 05 04 vO 

CO rH CO TH 

^l>OCOh 




^ rH —1 04 

T-H 04 04 

TH 04 TH 04 

04 T-H T-H 

04 th 04 04 

rH TH 04 



• 

HH 

^ TH rH Q O O 

th t-h 04 04 GO 

GO GO ^ ^ vO 

k-O CO CO l> t— 

C- CO CO 05 05 

(O’ O 1 tH rH TH 




r— 1 iH 





rH rH rH H rH 


Days. 

1-t 04 GO ^ ^Ol 

CO 4> CO 05 O 

th 04 CO t^h iO 

CO t- CO 05 O 

th 04 GO rt* kO 

co r- co 05 o 



T-H 

i-H t-h T-H T-H »-H 

H H - H 04 

04 04 04 04 04 

04 04 04 04 GO 




• 

CD 1> CO 05 

05 o O T-H 04 

04 CO ^ iO 

CO CO l" t- GO 

05 CT5 O th th 

04 GO GO rj« rjr 

kO 


c? 

^ Tf kO t-H 04 

CO kO t-h 04 

GO ^ lO T-H 

04 GO ^ iO 

TH 04 ^ kO 

th 04 GO ^ kO 




CO CO GO CO CO 

04 kO 05 04 kO 

CO T-H CO T-H 

^ t- O CO h 

O GO CO 05 GO 

CO 05 04 kO CO 

04 


4-> 

o 

" T-1 r-H 04 04 04 

GO GO GO ^ ^ 

O lO 

rH T-H t-H 

04 04 04 04 GO 

co co ^ ^ ^ 

kO 



o th r-n th th th 

T-H r-H T-H t-h t-h 

T-H T-H T-H TH 04 

04 04 04 04 04 

04 04 04 04 04 

04 04 04 04 04 

04 



r-H T*—< TH r-i rH 

f-H T-H t-H r-H T-H 

T-H t-h *V— ( T-H T-H 

T-H T-H T-H rH t-H 

rH rH rH rH tH 

r-H rH tH rH rH 

TH 



- CO O O 

GO h h O C5 

GO !> T-H LO 05 

GO CO O ^ OO 
kO kG) Tt GO 

04 CO O rf CO 

04 CO O ^ 05 

04 


• 

>> 

IS 

g 

GO 04 04 th 

lO ^ GO 

GO 04 04 t-h 

GO 04 04 th 

kO kO rf GO 

GO 


O ^ CO 04 CO 

OGO^thlO 

05 GO h th iO 

05 04 CD O rf 

CO 04 CO O *t 

CO th kO 05 GO 

!> 

~*n 

Zfl 

^ lO iO kO 

T-H T-H T-H G4 04 

04 GO GO ^ ^ 

^ kQ kG 

rH rH 04 04 

04 CO CO CO ^ 


P 

bD 

P 

H 

o 

. 04 to CO 04 lO 

CO TH -t h o 

CO CO 05 04 id 

CO t-h rf CO t-h 

rf t- O GO CO 

05 04 k-O CO th 

rf 

H 

° 04 rH rH 

04 T-H 04 04 

t-h 04 t-h 04 

04 t-h 

th 04 th 04 

TH TH 04 tH 

04 


C/5 CO CO GO O O 

05 t-h t-h 

t-h t-h rH 04 04 

GO CO rf vO 

kO kO (O CO 

1> CO CO CO 05 

05 



rH rH 

T^ T-H T-H 







• 

^ l-H CO T-H CO Hi 

CO T-H CO T-H CO 

T-H CO T-H CO T-H 

CO rH J> 04 D* 

04 1> 04 i> 04 

!> 04 1> 04 1> 

04 


A 

04 k<0 GO ^ 

H o 04 CO 

T-H 04 lO CO 

T-H kO 04 

GO TH Tf 04 

kO CO Tt H 

kO 


a 

^ rf rf iiC CO CO 

t'* D- CO 05 05 

O O t-h t-h 04 

CO CO ^ ^ l5 

CO CO L* CO 

CO 05 o O TH 

tH 


■4-> 

CO lO t-H 

04 GO ^ ^5 

04 GO ^ lQ 

t-h 04 GO rf kT5 

th 04 GO r* 

kO 04 GO ^ 

kO 


o 

n CO CO 04 CO CO 

04 CO t-h lO 

CO T-H t-h 

1> O GO CO 

O GO CO 05 04 

kO 05 04 kO CO 

H 



° H 04 H GJ 

04 »H TH 

04 hh 04 04 

T-H T-H 04 

th 04 TH 

th 04 t-h 04 

04 


s 

CO CO 05 O O TH 

h O hh t-h t-h 

th 04 04 GO GO 

r}r cO iO kO 

CO CO 1> 1> CO 

CO CO 05 05 O 

o 


r-H i-H t-H 

T-H 




tH 

tH 

Days. 

t-h 04 CO ^ iO 

CO lr- CO 05 O 

h 04 GO ^ ^5 

CO t- CO O O 

H G) CO Tt kC 

CD l' CO 05 O 

H 


T-H 

r—i r-H T—i ^-H t-H 

-h t-h t-h t-h 04 

04 04 04 04 04 

04 04 04 04 GO 

GO 




- CO CO CO 

05 O t-h t-h 04 

GO GO t^t rjr lO 

CO- CD i> CO CO 

05 O O th 04 

04 GO GO ^ kQ 

kO 


cS 

t-h 04 GO ^ kO 

04 CO ^ kO 

t-h 04 GO ^ 

kTO T-H 04 GO 

rt< th 04 GO 

^ kO th 04 

GO 



^ CO h ^ jr O 

t- O CO CO 

O GO CO 05 04 

kO 05 04 kO CO 

th kO 00 h rt< 


GO 


■*-> 

o 

GO ^ ^ ^ & 

kT5 kD 

T-H T-H l-H T-H 04 

04 04 GO GO CO 

Tt Tf kO kO 

kO rH 



s 

o D D D O D 

05 05 o O O 

o o o o o 

O O O O O 

o o o o o 

O TH tH rH rH 

th 




T-H T-H T-H 

T-H T-H T-H T-H T-H 

rH t-H rH t-h t-h 

rH th tH tH rH 

tH rH rH H rH 

rH 



^ o ^ CO 04 CD 

O T-H CO 04 CO 

O CO h co 

O ^ CO 04 CD 

O ^ t -- kO 

C5 GO Tt H kO 

C5 


• 

' Tt GO 04 04 t-h 

T-H kQ iO TCf 

^ GO 04 04 t-h 

t-h kO uO rf 

Tt GO 04 04 TH 

kO kO TjT 

GO 



05 GO r- t-h iO 

05 CO CO o ^ 

CO 04 CO O rh 

CO 04 kO 05 CO 

t— t-h kO 05 GO 

-r-H 00 

CO 


a 

a 

o 

rt iO uO 

t-h h 04 04 

04 GO GO rf ^ 

Tt kO kO kO 

rH tH rH 04 

04 GO GO CO ^ 



O i> O GO t'* O 

GO CO 05 04 lO 

CO T-H rf 1> O 

GO CO 05 04 CO 

05 04 kO CO th 

^ I> O GO CO 

05 

P 

r* 

04 t-h 

»-H 04 04 

T-H T-H 04 T-H 

04 ^ th 

04 th 04 04 

TH TH 04 



<1 

CZ5 !> CO CO 05 

05 05 O O th 

H O O O T-H 

t-h 04 04 CO GO 

GO rf tF kO kO 

CO CO l> 

co 




T-H T-H T-H 

T-H 






• 

oooo^o 

O lO o o 

lO O t-h CO 

rH CD t-h CD t-H 

CO h CO th CO 

TH CO TH CO TH 

CO 


A 

" T-H LO 04 GO 

T-H ^ iT5 go 

rf t-h i-O 04 

GO t-h •Hf 04 

kO GO rT th 

kO 04 GO TH 



a 

CD CO L- CO CO 

05 05 O O t-h 

04 04 GO GO Tl< 

kO kO CO CO U- 

r- co 05 05 O 

O TH 04 04 GO 

GO 


o 

" th 04 CO ^ 

UO 04 GO rf 

iO t-h 04 GO 

tJt lQ t-h 04 

GO rf kO 04 

GO rf kO rH 

04 


O 

_ OD h ^ O 

GO t' O CO CO 

C5 GO CD 05 04 

kO CO 04 kO CO 

rH ^ l*- H ^ 

O GO £- O 

GO 



° 04 t-h 04 04 

T-H H 04 

04 t-h 

T-H 04 rH 04 

04 TH rH 

04 rH 04 04 



• 

CO CO CO 05 05 

<^> CD TH ^H tH 

O O t-h t-h 04 

04 04 GO GO rt* 

rf kO kO CO CO 

CO 1> 1> CO CO 

05 




H T-H t-H tH t-H 






Days. 

t-h 04 CO ^ iO 

CO CO 05 O 

rH 04 GO rf kO 

co r- 05 05 O | 

rH 04 GO ^ kO 

CO C' CO Q o 


T-H 

T-H T-H T-H T-H T-H 

rH rH rH t-H 04 

04 04 04 04 04 

04 04 04 04 GO 

GO 


/ 












































































































TABLE III. The Moon's Mean Motions for Months and Days. 


APPENDIX TO THE ASTRONOMY 



\ 


J 

; 


tj* 

iD 

iD 

CO 

CO 

P* 

CC 

CO 

CD 

O 

o 

T-H 

G* 

CD 

CO 

co 


ID 

iD 

CO 

CO 


CO 

05 

05 

O 

O 

Hi 

Gl 

Gl 

CO 

♦ 

c? 


GD 

CO 


ID 


T-H 

G< 

CO 



T-H 

G* 

CO 


iD 



CD 

CO 


iD 



Gl 

CO 

iD 


Hi 

Gf 

CO 

rr . 





P- 

O 

cg\ P- 

o 

CO 

co o 

CO 

CO 

CD 

G* 

iD 

00 

G* 

iD 

00 


-Hi 

{> 

f—^ 


L- 

o 

CO 

P* 

o 

CO 

CD 

05 | 


-h 

o 



Hji 

iD 

iD 

ID 








G* 

G* 

G* 

CO 

00 

CO 



—H 

iD 

iD 

iD 





fH 

tH 

TH 


HH 



P~ 

p- 

l- 

p- 

CO 

00 

00 

oo 

CO 

CO 

CO 

CO 

CC 

00 

CC 

CO 

CO 

00 

CO 

CO 

00 

CO 

CO 

05 

<05 

05 

05 

05 

05 

05 1 




Hi 

T-H 

T-H 

T-H 

T-H 

— 

T-H 


—•* 


T-H 




T-H 




Hi 

rH 

T—« 


TH 

T— 

tH 

Hi 

Hi 

TH 

Hi 

Hi 

Hi l 




CD 

o 


CO 

Gl 

CO 

o 

H+i 

CO 

G* 

CO 

o 


CO 

T-H 

iD 

05 

«co 

p- 

T—H 

iD 

05 

CO 

P- 

TH 

ID 

05 

CO 


Hi 

iD 


• 



T-H 


iD 

iD 



CO 

CD 

G* 


T-H 


ID 

ID 


CO 

CO' 

GJ 

G* 




iD 

iD 


CO 

CO 

Gl 

GD 

rH t 

• 

* 



CO 

o 

-i* 

!> 

T— 

iD 

CO 

CO 

P- 

T-H 

ID 

Cl 

CO 

re 

'-im* 

O 


00 

G* 

CO 

O 


CO 

Gl 

iD 

05 

co 

t- 

Hi 

iD 

05 

co : 

Q 

rt 

fl 



iD 

iD 

iD 




T-H 

T-H 

G* 

G* 

G* 

CO 

CO 




ID 

iD 




TH 

TH 

H 

Gl 

Gl 

CO 

CO 

CO 

^ i 

g 

C 

o 


CO 

05 

CD 

iD 

<o 

Gi 

iD 

CO 



J> 

/—s 

CO 

CO 

C5 

G* 

iD 

CO 

pJ 

iD 

CO 

TH 

rt- 

p- 

o 

CO 

CD 

05 

GD 

iD 

co 1 


c 

o 

gi 


Gl 


T-H 


T-H 

G* 

T—« 

G* 


G* 


T-H 

G* 

TH 

G* 


G* 


TH 


TH 

Gl 

TH 

Gl 


T-i 


Hi 

Gi ; 

o 

< 

, 
































V 


in 

Hi 

GD 

GD 

CO 

GO 




iD 

iD 

CO 

CO 

p- 

P- 

P* 

00 

CO 

05 

05 

O 

o 

TH 

HH 

TH 

o 

O 

tH 

Hi 

GD 

Gl 

GD [ 

Q 






















T-H 



TH 










• 


-H 

O 

iD 

O 

iD 

O 

iD 

o 

iD 

O 

iD 

o 

iD 

o 

iD 

o 

iD 

o 

iD 

o 

iD 

'■H 

ID 

o 

iD 

w 

iD 

o 

iD 

O 

iD | 



*** 

GO 

T—H 


Gl 

iD 

CO 



*+ 

iD 

G* 


CO 



G4 

iD 

co 




iD 

CD 


CO 

r “‘ 


GD 

iD 

CO 



(Pi 


iD 

CO 

CO 

l- 

P- 

CO 

a> 

CO 

O 

O 

PH 

G* 

G* 

CO 

CO 

-* 

TH 

iD 

CO 

CO 

P- 


CO 

05 

05 

o 

V 

Hi 

Hi 

Gl 

co ! 


• 0^4 

k. 


T-H 

Gl 

co 


iD 


T—H 

CO 

'n 1 

iD 



G* 

CO 


iD 



G* 

CO 


iD 


TH 

co 

N 

iD 


T- 

Gl ! 


O 


rt 


O 

CO 

CO 

CO 

co 

CO 

CD 

G^ 

iD 

CD 


iD 

CO 

TH 


CO 

T-H 

T^}H 

p- 

O 

CO 

P* 

o 

CO 

CD 

05 

CO 

CD 

05 | 


s 

o 


T-H 


T-* 

Gl 


G( 


T-H 



G* 


CD 


G* 


TH 


TH 

G* 

TH 

Gl 


GD 


rH 

Gl 

T “ l 

Gl 




in 

CO 

GO 


Tt 1 


iD 

iD 

CO 

CO 

P* 

if 

P- 

CO 

CO 

05 

C5 

O 

O 

T-H 

PH 

TH 

o 

q 

H 

TH 

GD 

Gl 

Gl 

CO 

co 

^ j 























*H 

*—< 










. 




T-H 

Gl 

CO 


iD 

CO 

P* 

00 

05 

O 

PH 

G* 

CO 


iD 

CO 


00 

05 

o 

TH 

CM 

CO 

-+ 

iD 

CO 

P- 

CO 

05 

O 

r-^ 












T—< 

' 



— 

*-< 

*—< 



TH 

CD 

GD 

Gl 

Gl 

CD 

Gl 

GD 

CD 

Gl 

GD 

co 

co i 

\ 




id 

iD 

CO 

p- 


00 

CO 

CD 

o 

o 

PH 

G* 

G* 

CO 


-H 

iD 

CO 

CO 

P* 

CO 

CO 

05 

05 

O 

Hi 

Hi 

Gl 

CO 

CO 

{ 


d 



T-H 

Gl 

CO 


iD 


T-H 

CO 


iD 



G* 

CO 


iD 


T-H 

G* 

co 


iD 


GD 

co 


iD 


H 

| 



CO 

Gl 

iD 

CO 

T-H 

rf 

CO 




O 


P- 

O 

CO 

CO 

05 

CO 

CO 

05 

G* 

iD 

CO 

Gl 

iD 

CO 

TH 


CO 

rH 



-+-j 

o 



T-H 

T-H 

T-H 

GD 

G^ 

G^ 

CO 

CO 

CO 




iD 

iD 

iD 

iD 




TH 


TH 

Gl 

GD 

Gl 

CO 

CO 

CO 




HH 


CO 

CO 

CO 

CO 

CO 

CO 

CO 

CO 

CO 

CO 

CO 

co 

CO 

CO 

CO 

CO 

CO 



!> 

p* 

P- 



!> 

P- 



P” 

P- 





Hi1 

Hi 

h< 

Hi 

T-H 

T—< 

T-H 

T-H 

t-H 


T-H 

1 

PH 

1 

T-H 

H 

TH 



T—H 

TH 

TH 

TH 

tH 

tH 

tH 

Hi 

TH 

tH 

Hi 

i 





T-H 

iD 

CO 

00 

p* 


ID 

CD 

CO 

P“ 

TH 

iD 

CD 

CO 

P* 

T— 

iD 

05 

GO 

CO 

o 

rr 

CO 

Gl 

CO 

o 

-fT 

CD 

Gl 



• 

** 

T-H 

T-« 


iD 

iD 



CO 

G* 

G^ 


T-H 


iD 

iD 



CO 

G* 

G* 

H 



iD 

iD 



co 

Gl 

CD 


C 



05 

GO 

1> 

O 


CO 

G) 

CO 

O 


CO 

G* 

CO 

CD 

CO 


T-H 

uO 

CD 

GO 

C- 

PH 

ID 

CO 

GD 

CO 

o 


CO 

Gl 


pO 

c3 

c 



vO 

iD 





T-H 

G* 

G* 

G* 

CO 

GO 

CO 



ID 

ID 

iD 



YH 

TH 

tH 

GD 

Gl 

CO 

CO 

CO 



a 

n 

o 



t- 

O 


P- 

o 

co 

CO 

CO 

G* 

iD 

CO 

TH 

rt 


Q 

CO 

co 

05 

CO 

CO 

05 

GD 

iD 

CO 

TH 

rt 

p 

O 

CO 



a 

o 

gi 


Gl 


T-H 


T-H 

G* 


G^ 


T-H 


T-H 

G* 


G* 


TH 


TH 

Gl 

TH 

GD 


GD 


Hi 


rH 


> 

<2 

































o 


m 

o 

th 

t-H 

Gl 

G* 

CO 

CO 

GO 



iD 

iD 

CD 

CO 

CO 


P* 

CO 

CO 

05 

05 

05 

o 

HH 

TH 

T-i 

O 


Hi 

Hi 


£ 

























TH 

TH 

TH 

Hi 







• 



CO 


CO 


CO 


CD 


CD 


05 


CD 

Tf 

05 


05 


05 


05 

H 

05 


05 


05 

rf 

05 






CO 



G^ 

iD 

CO 



T-H 

iD 

Gl 


CO 

T-H 


CD 

iD 

co 



r " H 

iD 

Gl 


CO 

TH 


Gl 

iD 





CO 

CO 

05 

CO 

O 

O 

T-H 

G* 

G* 

co 

CO 


iD 

iD 

CO 

CO 

t- 

P- 

CO 

05 

05 

o 

O 

tH 

GD 

Gl 

CO 

CO 








ID 


T-H 

CO 


iD 


T-H 

G* 

CO 


iD 


T-H 

G* 

CO 


iD 



CO 


iD 



Gl 

CO 

rp 

O 



o 


CO 

TH 

iD 

CO 




T-H 



o 

CO 

CO O 

CO 

CO 

05 

G* 

ID 

05 

G* 

iD 

CO 

rH 

iD 

CO 

Hi 

•+ 

P* 

o 




o 

GD 


GD 


G^ 




T-H 

G* 

T-H 

G* 


G* 


T-H 

G* 


G* 


CD 


TH 



GD 

TH 

Gl 


Gl 



• 

c/5 

T-H 

Gl 

GD 

CO 

CO 



iD 

ID 

iD 

CO 

CO 

P“ 

P* 

00 

CO 

CO 

05 

05 

o 

O 

TH 

H 

o 

a 

O 

H« 

PH 

Gl 

GD 
























H 

TH 

*-H 

TH 











T—H 

GD 

co 


iD 

CO 

t- 

00 

CO o 

TH 

G* 

CO 


ID 

CO 

p- 

CO 

05 

o 

TH 

Gl 

CO 


iD 

CD 


CO 

05 

O 


J J tlj O • 











T— 

T-H 

PH 

T-H 

TH 

H 

H 




CD . 

Cl 

GD 

GD 

GD 

GD 

GD 

GD 

Gl 

GD 

CO 






i-O 

CO 

CO 

c- 

P- 

CO 

CO 

CD 

O 

T-H 

T—T 

G* 

CO 

co 


iD 

iD 

CO 

CO 

P- 

CO 

CO 

05 

O 

O 

Hi 

CD 

GD 

CO 

CO 

Tf 


c? 


CO 


iD 



G* 

CO 



T-H 

G* 

CO 


iD 


H 

G* 

00 


iD 


TH 

GD 


iD 



Gl 

co 


iD 



o 

CO 

CO 

o 

CO 

CO 

CO 

G* 

CO 

CD 

G* 

iD 

CO 

TH 

iD 

CO 




O 

Tj« 

P* 

O 

CO 

CD 

O 

CO 

CD 

05 

Gl 

iD 


-h 

o 


CO 

CO 

CO 





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iD 

iD 




T—H 

TH 

T-H 

G* 

G* 

G* 

CO 

co 

CO 




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o 


rf 




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Tj* 

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ID 

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iD 

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iD 

iD 

iD 

iD 

iD 

iD 

iD 

iD 

CD 

CD 




H< 


Hi 

T-H 

T—4 

T-H 

T-H 

T-< 

▼— 1 

T-H 


*-* 


H 

—* 



TH 

T-H 

TH 

TH 

H 

TH 

TH 

rH 

T— ■« 

Hi 

Hi 

TH 

Hi 

Hi 




iD 

CO 

CO 

CO 

O 


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G* 

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CO 

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CO 

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o 

rf 

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GD 

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CO 

TH 

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05 

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GD 

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co 

CO 

G* 

T-H 




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CO 

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T— 1 

TH 



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rf 


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GD 

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CO 

GD 

CO 

o 


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CD 

co 

t- 

TH 

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G5 

co 

CO 

o 

t-H 

CO 

CD 

CO 

O 


CO 

GD 

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05 

CO 

P- 

Hi 

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u 

QJ 

s 


rr 

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T-H 



Gt 

G* 

CO 

CO 

CO 



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iD 

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TH 

TH 

TH 

Gl 

GD 

Gl 

CO 

co 



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o 

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CO 

G) 

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00 

PH 


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rH 



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CO 

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05 

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Hi 

o 

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o 


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T-H 


t-H 

G^ 

T-H 

G^ 


G>< 


T-H 

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t—H 

G* 


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Pi 


rH 

Gl 

TH 

o 

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• 

rn 


t-H 

O 

o 

T-T 

— < 

T-H 

G* 

G^ 

CO 

CO 




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CO 

CO 


c- 

CO 

CO 

CO 

05 

05 

o 

a 

Hi 

rH 


o 

w 



Hi 

T-H 
























Hi 

Hi 

Hi 

TH 

Hi 



• 

V. 

CO 

CO 

CO 

CO 

CO 

CO 

CO 

co 

CO 

CO 

CO 

CO 

CO 

co 

CO 

CO 

CO 

co 

CO 

CO 

CO 

CO 

CO 

CO 

CO 

CO 

CO 

T-3* 

05 

T^i 

05 




iD 

CO 



T-H 

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G< 


CO 

T—H 

rf 

G< 

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co 


rt 


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G* 


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TH 


Gl 

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CO 


rf 

T-i 

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GD 


O 

Q 


CO 

O 

T-H 

h, 

G* 

G^ 

CO 

-f 


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iD 

CO 

CO 


CO 

CO 

05 

CD 

O 

TH 


CD 

GD 

CO 

CO 


iD 

iD 

CD 

CD 

P* 


• ^ 


*-• 

CO 


iD 



G^ 

CO 


iD 



O'* 

C0 


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CO 


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H 

GD 

CO 


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TH 

GD 

CO 


o 


o 

co 

CO 

CO) 

CO 

CO 

C5 

G* 

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CO 

G* 

iD 

00 

T-H 

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P* 

TH 


p- o 

CO 

P* 

o 

CO 

CO 

05 

GD 

CD 

05 

Gl 

iD 


s 


T-H 

GD 


T-H 



04 

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G* 


G* 




T-H 

G* 


G* 


CD 


PH 


TH 

Gl 


GD 


TH 





in 

o 

O 

T-H 

T-H 

CD 

G^ 

o* 

CO 

CO 



iD 

iD 

CO 

CO 

CO 


P- 

CO 

CO 

05 

05 

o 

O 

o 

TH 

TH 

O 

o 

YH 

_4 


























pH 

-H 

TH 

th 

tH 





Dnvsi 


h4 -CD 

CO 


iD 

CO 

p- 

CO 

CD 

O 

H 

CD 

CO 


iD 

CO 


CO 

05 

o 

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GD 

co 


iD 

CD 

r* 

CO 

05 

o 
















T—H 



T-H 

T—H 

T-H 


GD 

Gl 

Gl 

Gl 

Gl 

GD 

GD 

Gl 

Gl 

GD 

CO 

CO , 


\ 












































































































LUNAR TABLES 


273 


TABLE IV. The Moon's Mean Motions for Hours , Minutes , 
and Seconds. 


H 

/ 

V 

M. 

O 

f 

Mo. 3. 

’ n 

" »/ 

•II It/t 

Anomaly. 

O I " 

t n m 

u m nn 

Mot. ft 

/ It 

/' HI 

if; lift 

/ 

ir 

M. 

/ 

it 

Mo. 

// 

It* 

J) • 

III 

//// 

Anomaly. 

/ a w 

a m m; 

Mot. ft. 

' " 

it in 

1 

0 

32 

56 

0 

32 

39 

0 

8 

31 

17 

1 

10 

16 

52 

32 

4 

6 

2 

1 

5 

53 

1 

5 

20 

0 

16 

32 

17 

34 

7 

17 

25 

13 

4 

14 

3 

1 

38 

49 

1 

37 

59 

0 

24 

33 

18 

7 

3 

17 

58 

0 

4 

22 

4 

2 

11 

46 

2 

10 

39 

0 

32 

34 

18 

40 

0 

18 

30 

31 

4 

30 

5 

2 

44 

42 

2 

43 

18 

0 

40 

35 

19 

12 

55 

19 

3 

12 

4 

37 

6 

3 

17 

39 

3 

15 

59 

0 

48 

36 

19 

45 

53 

19 

35 

50 

4 

47 

7 

3 

50 

35 

3 

48 

38 

0 

56 

37 

20 

18 

48 

20 

8 

30 

4 

54 

8 

4 

23 

32 

4 

21 

18 

1 

4 

38 

20 

51 

47 

20 

41 

10 

4 

54 

9 

4 

56 

2° 

4 

53 

58 

1 

12 

39 

21 

24 

41 

21 

13 

49 

5 

1 

10 

5 

29 

25 

5 

26 

38 

1 

19 

40 

21 

57 

37 

2l 

46 

30 

5 

12 

11 

6 

2 

21 

5 

59 

17 

1 

27 

41 

22 

30 

36 

22 

19 

12 

5 

25 

12 

6 

35 

18 

6 

31 

57 

1 

35 

42 

23 

3 

30 

22 

51 

48 

5 

31 

13 

7 

8 

14 

7 

4 

37 

1 

43 

43 

23 

36 

30 

23 

24 

30 

5 

42 

14 

7 

41 

10 

7 

37 

16 

1 

51 

44 

24 

9 

24 

23 

57 

6 

5 

50 

15 

8 

14 

7 

8 

9 

56 

1 

59 

45 

24 

42 

19 

24 

29 

48 

6 

0 

16 

8 

47 

3 

8 

42 

36 

2 

7 

46 

25 

15 

18 

25 

2 

30 

6 

6 

17 

9 

20 

0 

9 

15 

16 

2 

15 

47 

25 

48 

12 

25 

35 

6 

6 

13 

18 

9 

52 

56 

9 

47 

55 

2 

23 

48 

26 

21 

12 

26 

7 

48 

6 

23 

19 

10 

25 

53 

10 

20 

35 

2 

31 

49 

26 

54 

6 

26 

40 

30 

6 

30 

20 

10 

58 

49 

10 

53 

15 

2 

39 

50 

27 

27 

0 

27 

13 

6 

6 

36 

21 

11 

31 

46 

11 

25 

55 

2 

47 

51 

28 

0 

0 

27 

45 

48 

6 

48 

22 

12 

4 

42 

11 

58 

34 

2 

55 

52 

28 

32 

55 

28 

18 

30 

6 

54 

23 

12 

37 

39 

12 

31 

14 

3 

3 

53 

29 

5 

54 

28 

51 

6 

7 

*0 

24 

13 

10 

35 

13 

3 

54 

3 

11 

54 

29 

38 

48 

29 

23 

48 

7 

6 

25 

13 

43 

32 

13 

36 

34 

3 

19 

55 

30 

11 

48 

29 

56 

24 

7 

18 

26 

14 

16 

28 

14 

9 

13 

3 

26 

56 

30 

44 

42 

30 

29 

6 

7 

24 

27 

14 

49 

24 

14 

41 

53 

3 

34 

57 

31 

17 

36 

31 

1 

48 

7 

30 

28 

15 

22 

21 

15 

14 

33 

? 

42 

58 

31 

50 

36 

31 

34 

24 

7 

48 

29 

15 

55 

17 

15 

47 

13 

3 

50 

59 32 

23 

30 

32 

7 

6 

7 

48 

30 

16 

28 

14 

16 

19 

52 

3 

58 

60(32 

56 

39 

32 

39 

48 

7 

56 


37J 


TABLES of the MOON’S EQUATIONS. 


I. Annual Equation of the Moon's Node, 
i 

Argument. Sun’s Mean Anomaly. 


S. 

+ 

0 

4- 

1 

+ 

2 

+ 

3 

+ 4 

+ 

5 

S. 

o 


II 

/ 

tl 

t 

n 

/ 

It 

I 

II 

/ 

ll 

0 

0 

0 

0 

4 

30 

7 

52 

9 

12 

8 

4 

4 

42 

30 

50 

47 

5 

10 

8 

15 

9 

11 

7 

38 

3 

58 

25 

10 

1 

34 

5 

48 

8 

34 

9 

6 

7 

9 

3 

13 

20 

15 

2 

19 

6 

23 

8 

49 

8 

56 

6 

37 

2 

26 

15 

20 

3 

4 

6 

56 

9 

1 

8 

43 

6 

1 

1 

38 

10 

25 

3 

48 

7 

26 

9 

8 

8 25 

5 

23 

0 

49 

5 

30 

4 

30 

7 

52 

9 

12 

8 

4 

4 

42 

0 

0 

0 

S. 


-11 


-10 


-9 


-8 


-7 


-6 

S. 


Annual Equation of Moon's Mean Anomaly. 


Argument. Sun's Mean Anomaly. 


s. 

+ 

0 

+ 

1 

+ 

2 

+ 

3 

+ 4 

+ 

5 

S.! 

o 

l 

II 

/ 

ll 

/ 

II 

> 

II 

/ 

II 

• 

// 

o J 

0 

0 

0 

10 

37 

18 

33 

21 

42 19 

1 

11 

5 

30 

5 

1 

51 

12 

11 

19 

27 

21 

39 18 

1 

9 

23 

25 

10 

3 

40 

13 

41 

20 

13 

21 

27 

16 

53 

7 

36 

201 

15 

5 

29 

15 

4 

20 

49 

21 

5 

15 

37 

5 

45 

15 

20 

7 

15 

16 

21 

21 

16 

20 

34 

14 

13 

3 

52 

10 

25 

8 

58 

17 

31 

21 

34 

19 

52 

12 

42 

1 

56 

5 

30 

10 

37 

18 

33 

21 

42 

19 

1 

11 

5 

0 

0 

0 

S. 


-11 


-10 


-9 


—8 

—7 


-6 

s. 


I. For the Moon's Longitude. 
Argument. Sun’s Mean Anomaly. 


s. 

+ 

0 

+ 1 

-f 2 

+ 

3 

+ 

4 

+ 

5 

S. 

0 

' 

// 

/ 

" 

1 

" 

1 

// 

' 

ll 

' 

ll 

o 

0 

0 

0 

5 

27 

9 

31 

11 

9 

9 

47 

5 

42 

30 

5 

0 

57 

6 

15 

9 

59 

11 

8 

9 

16 

4 

49 

25 

10 

1 

53 

7 

1 

10 

23 

11 

1 

8 

41 

3 

54 

20 

15 

2 

49 

7 

44 

10 

41 

10 

50 

8 

2 

2 

58 

15 

20 

3 

43 

8 

23 

10 

55 

10 

34 

7 

19 

1 

59 

10 

25 

4 

36 

8 

59 

11 

4 

10 

13 

6 

32 

1 

0 

5 

30 

5 

27 

9 

31 

11 

9 

9 

47 

5 

42 

0 

0 

0 

S. 

—11 


-10 


-9 


-8 


-7 


-6 

s. 





















































































274 


APPENDIX TO THE ASTRONOMY. 


II. For the Mooli's 
Longitude. 

O 


III. For the Moon's 
Longitude. 


V. For the Moon’s Longitude. Evection. 


Arg. II. 2 D from © 
+ Arg. 1. 


Arg. III. 2 D from © 
— Arg. I. 


Argument V. 

2 1) from © — j) ’s Mean Anomaly. 


"s. 

— 0 

— 1 

— 2 

S. 

s. 

-f 6 

+ l\+ 8 

S. 

o 

// 

// 

II 

0 

0 

0 

28 

48 

30 

5 

5 

32 

51 

25 

10 

10 

36 

52 

20 

15 

14 

39 

54 

15 

20 

19 

43 

55 

10 

25 

24 

46 

56 

5 

30 

28 

48 

56 

0 

S. 

■f 11 

+ 10 

+ 9 

S. 

S. 

—5 

—4 

—3 

s. 


s. 

— 0 

— 1 

— 2 

S. 

s. 

+ 6 

+ 7 

+ 8 

S. 

o 

/ // 

/ // 

/ II 

o 

0 

0 0 

0 38 

1 5 

30 

5 

0 7 

0 43 

1 8 

25 

10 

0 13 

0 48 

1 11 

20 

15 

0 19 

0 53 

1 13 

15 

20 

0 26 

0 58 

1 14 

10 

25 

0 32 

1 2 

1 15 

5 

30 

0 38 

1 5 

1 15 

0 

S. 

->-n 

+ 10 

+ 9 

S. 

s. 

—5 

—4 

—3 

s. 


IV. For the Moon’s Longitude. 

Arg. IV. 2 3 from © + 3 ’s Mean Anomaly. 


s. 

+ 0 

+ 1 

+ 2 

S. 

s. 

—6 

—7 

—8 

S. 

o 

II 

II 

If 

o 

0 

0 

29 

50 

30 

5 

5 

33 

52 

25 

10 

10 

37 

54 

20 

15 

15 

41 

56 

15 

20 

20 

44 

57 

10 

25 

24 

47 

57 

5 

30 

29 

50 

58 

0 

S. 

—11 

—10 

— 9 

S. 

s. 

+ & 

+ 4 

+ 3 

s. 


s. 


— 

0 


— 

1 


— 

2 


— 

3 


— 

4 


— 

5 

|S. 

o 

o 

/ 

n 

O 

/ 

// 

O 

I 

ll 

0 

1 

II 

O 

t 

It 

o 

/ 

" 

1 0 

0 

0 

0 

0 

0 

39 

44 

1 

9 

11 

1 

20 

28 

1 

10 

120 

40 

45 30 

1 

0 

1 

23 

0 

40 

56 

1 

9 

53 

1 

20 

29 

1 

9 

30 0 

39 

31 29 

2 

0 

2 

46 

0 

42 

7 

1 

10 

34 

1 

20 

28 

1 

8 

46 0 

38 

16128 

3 

0 

4 

9 

0 

43 

18 

1 

11 

14 

1 

20 

25 

1 

8 

10 

37 

0 27 

4 

0 

5 

32 

0 

44 

27 

1 

11 

52 

1 

20 

21 

1 

7 

15 0 

35 

45 

26 

5 

0 

6 

55 

0 

45 

36 

1 

12 

CD 1 

1 

20 

16 

1 

6 

28 0 

34 

27 

25 

6 

0 

8 

17 

0 

46 

45 

1 

13 

5 

1 

20 

9 

1 

5 

400 

33 

10 

24 

7 

0 

9 

40 

0 

47 

52 

1 

13 

39 

1 

20 

1 

1 

4 

500 

31 

52 

23| 

8 

0 

11 

20 

48 

59 

1 

14 

12 

1 

19 

51 

1 

3 

59 

0 

30 

33 

22 

9 

0 

12 

24 

0 

50 

4 

1 

14 

44 

1 

19 

40 

1 

3 

7 

0 

29 

14 

21 

10 

0 

13 

46 

0 

51 

9 

1 

15 

15 

1 

19 

27 

1 

2 

13 

0 

27 

54 

20 

11 

0 

15 

8 

0 

52 

13 

1 

15 

44 

1 

19 

13 

1 

1 

19 

0 

26 

34 

19 

12 

0 

16 

30 

0 

53 

16 

1 

16 

11 

1 

18 

57 

1 

0 

23 

0 

25 

13 

18 

13 

0 

17 

51 

0 

54 

18 

1 

16 

38 

1 

18 

40 

0 

59 

26 

0 

23 

51 

17 

14 

0 

19 

12 

0 

55 

19 

1 

17 

3 

1 

18 

21 

0 

58 

28 

0 

22 

29 

16 

15 

0 

20 

32 

0 

56 

19 

1 

17 

26 

1 

18 

1 

0 

57 

29 

0 

21 

7 

15 

16 

0 

21 

52 

0 

57 

18 

1 

17 

48 

1 

17 

40 

0 

56 

29 

0 

19 

44 

14 

17 

0 

23 

12 

0 

58 

16 

1 

18 

9 

1 

17 

17 

0 

55 

28 

0 

18 

22 

13 

18 

0 

24 

31 

0 

59 

13 

1 

18 

29 

1 

16 

53 

0 

54 

26 

0 

16 

58 

12 

19 

0 

25 

50 

1 

0 

9 

1 

18 

47 

1 

16 

27 

0 

53 

22 

0 

15 

34 11 

20 

0 

27 

9 

1 

1 

4 

1 

19 

3 

1 

16 

0 

0 

52 

18 

0 

14 

10 10 

21 

0 

28 

27 

1 

1 

58 

1 

19 

18 

1 

15 

31 

0 

51 

13 

0 

12 

46 

9 

22 

0 

29 

44 

1 

2 

51 

1 

19 

32 

1 

15 

1 

0 

50 

7|0 

11 

22 

8 

23 

0 

31 

1 

1 

3 

42 

1 

19 

44 

1 

14 

30 

0 

49 

00 

9 

57 

7 

24 

0 

32 

18 

1 

4 

33 

1 

19 

55 

1 

13 

57 

0 

47 

51 0 

8 32 

6 

25 

0 

33 

34 

1 

5 

22 

1 

20 

4 

1 

13 

23 

0 

46 

42 

0 

7 

7 

5 

26 

0 

34 

49 

1 

6 

10 

1 

20 

12 

1 

12 

47 

0 

45 

33 

0 

5 

42 

4 

27 

0 

36 

4 

1 

6 

57 

1 

20 

18 

1 

12 

10 

0 

44 

22 

0 

4 

16 

3 

28 

0 

37 

18 

1 

7 

43 

1 

20 

23 

1 

11 

32 

0 

43 

10 

0 

2 

51 

2 

29 

0 

38 

31 

1 

8 

28 

1 

20 

26 

1 

10 

53 

0 

41 

58 

0 

1 

25 

1 

30 

0 

39 

44 

1 

9 

11 

1 

20 

28 

1 

10 

12 

0 

40 

450 

0 

0 

0 

S. 

+ 11 

+ 10 

+ 9 


+ 

8 


+ 

7 

+ 6 j 

v 















































































*■ 


"V 1. For the 5 ’i Longitude. 
Argument. 


Arg. V. + Arg. I. 


s. 

+ 

0 

+ 

1 

+ 

2 

S. 

s. 

— 6 


7 

— 

8 

s. 

o 

t 

n 

! 

ft 

f 

// 

o 

1 

0 

0 

1 

2 

1 

47 

30 

5 

0 

11 

1 

11 

1 

52 

25 

10 

0 

21 

1 

19 

1 

56 

20 

15 

0 

32 

1 

27 

1 

59 

15 

20 

0 

42 

1 

35 

2 

2 

10 

25 

0 

52 

1 

41 

2 

3 

5 

30 

1 

2 

1 

47 

2 

3 

0 

S. 


11 


10 


9 

S. 

s. 

+ 

5 

+ 

4 

+ 

3 

s. 


LUNAR TABLES. 

VII. For the J 's Longitude. 


Argument. 



Arg. 

V. — 

Arg. 

I. 

s. 

+ 0 

+ 1 

+ 2 

S. 

s. 

— 6 

— 7 

— 8 

s. 

0 

// 

/' 

// 

0 

0 

0 

23 

40 

30 

5 

4 

27 

42 

25 

10 

8 

30 

44 

20 

15 

12 

33 

45 

15 

20 

16 

36 

46 

10 

25 

20 

38 

46 

5 

30 

23 

40 

47 

0 

S. 

— 11 

—10 

— 9 

S. 

s. 

+ 5 

+ 4 

+ 3 

s. 


275 

VIII. For the ])’s Longitude, 
.Argument. 

J)’s Mean Anom. — Arg. 1. 


s. 

+ 0 

+ 1 

+ ^ 

S. 

s. 

— 6 

— 7 

— 8 

5 . 

o 

// 

// 

" 

O 

0 

0 

21 

36 

30 

5 

4 

24 

38 

25 

10 

7 

27 

40 

20 

15 

11 

30 

41 

15 

20 

14 

32 

41 

10 

25 

18 

34 

42 

5 

30 

21 

36 

42 

0 

S. 

—11 

—10 

— 9 

S. 

s. 

+ 5 

4- 4 

+ 3 

s. 


IX. For the Moon's Longitude. 
Argument IX. 

Mean Dist. 5 from © — D’s Mean Anom. 


s. 

— 0 

— 1 

=F 2 

+ 3 

+ 4 

+ & 

S. 

o 

1 If 

/ It 

t H 

t it 

/ " 

t " 

0 

0 

0 0 

0 38 

0 30 

0 23 

1 9 

1 1 

30 

5 

8 

41 

23 

33 

1 12 

0 54 

25 

10 

16 

42 

16 

42 

1 14 

0 45 

20 

15 

23 

41 

7 

51 

1 13 

0 35 

15 

20 

29 

39 

+ 3 

58 

1 11 

23 

10 

25 

34 

35 

13 

1 5 

1 7 

12 

5 

30 

38 

30 

23 

1 9 

1 1 

0 0 

0 

S. 

+ 11 

+ 10 

d= 9 

— 8 

— 7 

— 6 

S. 


X. For the Moon's Longitude. 
Argument X. 

Mean Long. ft. — True Long. ©. 


s. 

+ 0 

+ 1 

+ 2 ; S. 

s. 

+ 6 

+ 7 

cri 

CO i 

+ 

o 

// 

/ n 

" 1 0 

0 

0 

0 52 

52 30 

5 

10 

0 57 

46 25 

10 

21 

0 59 

39 20 

15 

30 

1 0 

30 15 

20 

39 

1 0 

21 10 

25 

46 

0 57 

10 1 5 

30 

52 

0 52 

0 0 

S. 

—11 

—10 

— 9S. 

s. 

— 5 

— 4 

— 3S. 
































































276 


APPENDIX TO THE ASTRONOMY. 


XI. For the Moon's Longitude. Equation of the Centre. 


Argument IX. Moon's Correct Anomaly. 


s. 

— 0 


— 

1 


— 

2 

1 - 3 


— 

4 


— 

5 

Sj 

o 

0 



o 

/ 

// 

0 

/ 

// 

o 

/ 

// 

0 

/ 

// 

0 

/ 

// 

o 

0 

0 

0 

0 

2 

58 

30 

5 

16 

21 

6 

17 

38 

5 

38 

46 

3 

20 

56 

30 

1 

0 

6 

11 

3 

3 

59 

5 

19 

48 

6 

18 

2 

5 

35 

41 

3 

14 

57 

29 

2 

0 

12 

22 

3 

9 

24 

5 

23 

10 

6 

18 

18 5 

32 

28 

3 

8 

54 

28 

3 

0 

18 

32 

3 

14 

46 

5 

26 

26 

6 

18 

28 

5 

29 

8 

3 

2 

47 

27 

4 

0 

24 

42 

3 

20 

5 

5 

29 

38 

6 

18 

32 

5 

25 

42 

2 

56 

36 

26 

5 

0 

30 

52 

3 

25 

22 

5 

32 

43 

6 

18 

28 

5 

22 

10 

2 

50 

21 

25 

6 

0 

37 

2 

3 

30 

35 

5 

35 

43 

6 

18 

17 

5 

18 

30 

2 

44 

2 

24 

7 

0 

43 

11 

3 

35 

45 

5 

38 

38 

6 

17 

59 

5 

14 

44 

2 

37 

40 

23 

8 

0 

49 

19 

3 

40 

51 

5 

41 

27 

6 

17 

34 

5 

10 

52 

2 

31 

14 

22 

9 

0 

55 

26 

3 

45 

54 

5 

44 

11 

6 

17 

3 

5 

6 

53 

2 

24 

45 

21 

10 

1 

1 

33 

3 

50 

54 

5 

46 

48 

6 

16 

24 

5 

2 

48 

2 

18 

13 

20 

11 

1 

7 

39 

3 

55 

50 

5 

49 

20 

6 

15 

38 

4 

58 

37 

2 

11 

37 

19 

12 

1 

13 

44 

4 

0 

42 

5 

51 

45 

6 

14 

45 

4 

54 

19 

2 

4 

59 

18 

13 

1 

19 

47 

4 

5 

30 

5 

54 

5 

6 

13 

45 

4 

49 

56 

1 

58 

18 

17 

14 

1 

25 

50 

4 

10 

15 

5 

56 

19 

6 

12 

38 

4 

45 

25 

1 

51 

34 

16 

15 

1 

31 

51 4 

14 

56 

5 

58 

27 

6 

11 

244 

40 

50 

1 

44 

48 

15 

16 

1 

37 

51 4 

19 

33 

6 

0 

28 

6 

10 

34 

36 

8 

1 

37 

59 

14 

17 

1 

43 

494 

24 

5 

6 

2 

24 

6 

8 

34 

4 

31 

20 

1 

31 

8 

13 

18 

1 

49 

45 4 

28 34 

6 

4 

13 

6 

6 

59 

4 

26 

27 

1 

24 

16 

12 

19 

1 

55 

41 4 

32 

58 

6 

5 

56 

6 

5 

16 

4 

21 

29 

1 

17 

21 

11 

20 

2 

1 

344 

37 

18 

6 

7 

32 

6 

3 

27 

4 

16 

24 

1 

10 25 

10 

21 

2 

7 

254 

41 

33 

6 

9 

2 

6 

1 

30 

4 

11 

14 

1 

3 

27 

9 

22 

2 

13 

1514 

45 

44 

6 

10 

26 

5 

59 

27 

4 

5 

59 

0 

56 

27 

8 

23 

2 

19 

24 

49 

50 

6 

11 

435 

57 

16 

4 

0 

38 

0 

49 

27 

7 

24 

2 

24 

484 

53 

52 

6 

12 

54 5 54 

58 3 55 

13 

0 

42 

25 

6 

25 

2 

30 

31 4 

57 

49 

6 

13 

58 5 

52 

34 3 49 

42 

0 

35 

22 

5 

26 

2 

36 

125 

1 

41 

6 

14 

55 5 

50 

23 

44 

6 

0 28 

19 

4 

27 

2 

41 

505 

5 

29 

6 

15 

46 5 

47 

23 3 

38 

26 

0 

21 

15 

3 

28 

2 

47 

26j5 

9 

11 

6 

16 

30 5 

44 

383 

32 

40 

0 

14 

10 

2 

29 

2 

53 

05 

12 49 

6 

17 

7 i 5 

41 

46 3 

26 

51 

0 

7 

5 

1 

30 

2 

58 

1 * i 

1 o I 

1 <« 1 

16 

21 

6 

17 

38 5 

38 

463 

1 

20 

56 

c 

1 0 

0 

0 

S. 


f n 1 


f io 


+ 

9 1 


+ 8 l 


+ ■ 

7 

+ 6 

s. 


\ 



























LUNAR TABLES. 


XII. For the j) 's Long. Variation. 
Argument XII. 

3)’s equated Long-. — q's true Long. 


XIII. For the j)’s Long. 

Argument XIII. 

Doub. eq. dist. D from f l. 
— J C. An. 


_ 

+ 

0 

+ 1 


2 


3 


4 

r 

5 

S. 

0 

' 

II 

* 

II 

f 

n 

1 

ft 

/ 

ll 



0 

0 

0 

0 

30 

9 

29 

6 

2 

2 

32 

28 

31 

55 

31 

1 

1 

14 

30 

43 

28 

26 

3 

16 33 

3 

31 

15 

2S 

2 

2 

27 

31 

15 

27 

44 

4 

3033 

36 

30 

33 

28 

3 

3 

40 

31 

44 

27 

0 

5 

43 

34 

6 

29 

48 

27 

4 

4 

53 

32 

11 

26 

13 

6 

57 

34 

35 

29 

2 

26 

5 

6 

6 

32 

36 

25 

25 

8 

10 

35 

1 

,28 

13 

25 

6 

7 

18 

32 

58 

24 

35 

9 

22 

35 

24 

27 

22 

24 

7 

8 

30 

33 

18 

23 

43 

10 

34 

35 

45 

26 

29 

23 

8 

9 

41 

33 

35 

22 

49 

11 

46 

36 

3 

25 

35 

22 

9 

10 

51 

33 

49 

21 

53 

12 

56 

36 

19 

24 

38 

21 

10 

12 

0 

34 

1 

20 

56 

14 

6 

36 

32 

23 

40 

20 

11 

13 

9 

34 

11 

19 

57 

15 

15 

36 

43 

22 

40 

19 

12 

14 

16 

34 

18 

18 

57 

16 

23 

36 

51 

21 

39 

18 

13 

15 

23 

34 

22 

17 

55 

17 

29 

36 

57 

20 

35 17 

14 

16 

28 

34 

23 

16 

52 

18 

35 

36 

59 

19 

30 16 

15 

17 

31 

34 

22 

15 

47 

19 

39 

37 

0 

18 

24 15 

16 

18 

34 

34 

19 14 

41 

20 

42 

36 

57 

17 

17 14 

17 

19 

35 

34 

13 13 

34 

21 

44 

36 

53 

16 

8113 

18 

20 

35 

34 

4 

12 

26 

22 

44 

36 

45 

14 

58 12 

19 

21 

33 

33 

53 

11 

17 

23 

43 

36 

35 

13 

4711 

20 

22 

30 

33 

39 

10 

8 

24 

40 

36 

22 

12 

36 10 

21 

23 

24 

33 

23 

8 

57 

25 

35 

36 

6 

11 

23 

9 

22 

24 

17 

33 

4 

7 

45 

26 

29 

35 

48 

10 

9 

8 

23 

25 

8 

32 

43 

6 

34 27 

21 

35 

28 

8 

54 

7 

24 

25 

57 

32 

19 

5 

21 

28 

10 

35 

5 

7 

39 

6 

25 

26 

45 

31 

53 

4 

8 

28 

59 

34 

39 

6 

24 

5 

26 

27 

30 

31 

24 

2 

54 

29 

45 

34 

11 

5 

7 

4 

27 

28 

13 

30 

53 

'1 

4 t 

30 

29 

33 

41 

3 

51 

3 

28 

28 

54 

30 

19+0 

27 

31 

10 

33 

8 

2 

34 

2 

29 

29 

32 

29 

44 

0 

47 

31 

50 

32 

32 

1 

17 

1 

30 

30 

9 

29 

6 

—2 

2 

32 

CD I 

1 G-* 

31 

55 

0 

0 

0 

's. 


11 


10 

=F 

9 

+ 8 

+ 

7 

+ 

6 1 

s. 


s. 

+ 

+ 1 

+ 2 

3. 

s. 

- 6 

— 7 

— 8 

3. 

0 

/ li 

/ n 

t /• 

O 

0 

0 0 

0 42 

1 15 

’.0 

5 

0 7 

0 48 

1 16 

5 

10 

0 15 

0 54 

1 19 

20 

15 

0 22 

0 59 

1 21 

1 5 

20 

0 29 

1 4 

1 25 

10 

25 

0 35 

1 9 

1 24 

5 

30 

0 42 

1 13 

1 24 

0 

S. 

—11 

— 10 

— 9 

S. 

s. 

+ 5 

+ 4 

+ 3 

s. 


Equation of the Equinoctial 
Points. 

Argument. 

Mean Long. Moon’s 


s. 

— 0 

- 1 

— 2 S.' 

1 

s. 

+ 6 

+ "1 

CO 

+ 

S. 

0 

n 

" 

II 

O 

0 

0 

9 

16 

30 

5 

1 

10 

16 

25 

10 

3 

12 

17 

20 

15 

5 

13 

17 

15 

20 

6 

14 

18 

10 

25 

8 

15 

18 

5 

30 

9 

16 

18 

0 

3. 

+ 11 

+ 10 

+ 9 

S. 

,3. 

—5 

—4 

—3 

s 


38i 


277 

XIV. For the 3) ' s Long. Red. 
Argument. XIV. 

Long. D in Orb. — C. 
Long. ft. 


S. — 0 


s. 


00 

10 

20 
30 
0 


5 1 11 6 23 5 12 

6 1 25 

7 1 39 

8 1 52 

9 ! 2 6 


15 3 24 

16 3 36 
173 48 


20 


4 0 
4 1 


22 


4 

21 4 33 

22 4 43 

23 4 53 

24 5 


25 5 

26 5 21 

27 5 

28 5 
295 


126 


30 6 
38 6 


46 6 


2 3 . 


— 8 


5 53 5 53 

6 0 5 46 

6 6 5 38 

6 12 5 30 
6 18 5 21 


6 28 

5 3 

6 32 

4 53 

6 36 

4 43 

6 39 

4 33 

6 41 

4 22 

6 44 

4 11 

6 45 

4 *0 

6 47 3 48 


102 19 

11 2 33 

12 2 46 

13 2 59 

14'3 11 6 47 3 36 


6 48 3 24 
6 47i3 11 
47 2 59 
6 45 2 46 12 
44 2 33 11 


1 6 


6 41 2 19 10 


6 39 2 6 

6 36 1 52 
6 321 39 
6 281 25 


23 1 11 5 

6 180 57 4 
120 43 3 
6 0 28 2 
00 14 1 


30 5 53 5 53 0 0 0 


S. 


+ 11 


S. + 5 


+ 10 + 9 S. 


+ 4: + 3 S. 















































































































578 


APPENDIX TO THE ASTRONOMY, 


TABLES FOR FINDING THE MOON'S LATITUDE. 


I. For the .Moon's Latitude. 


II. For the .Moon's Latitude. 


Argument I. 


Argument II. 


Long. 3) in orbit— Cor. Long, £1 Double Dist. 3) in Orb. from © — Arg. I. 


s. • 

+ o 


+ 

1 


+ 

2 

s. 

s. ; 


— 

6 


— 

7 



8 

s. 

o ' 

o 

/ 

// 

0 

/ 

// 

O 

/ 

" 

o 

00 

0 

0 

2 

34 

18 

4 

27 

23 

30 

1 10 

5 

23 

2 

38 

56 

4 

30 

Q 

29 

_o 

0 

10 

46 

2 

43 

32 

4 

32 

37 

28 

30 

16 

9 

2 

48 

5 

4 

35 

6 

27 

40 

21 

31 

2 

52 

35 

4 

37 

31 26 

5 

0 

26 

53 

2 

57 

1 

4 

39 

50 25 

60 

32 

15 

3 

1 

24 

4 

42 

4 24 

7 

0 

37 

36 

3 

5 

44 

4 

44 

14 23 

80 

42 

56 

3 

10 

1 

4 

46 

17 

22 

90 

48 

16 

3 

14 

14 

4 

48 

16 

21 

100 

53 

35 

3 

18 

23 

4 

50 

9 

20 

110 

58 

52 

3 

22 

29 

4 

51 

58 

19 

12,1 

4 

9 

3 

26 

32 

4 

53 

40 

18 

13 

1 

9 

24 

3 

30 

30 

4 

55 

18 

17 

141 

14 

39 

3 

34 

25 

4 

56 

50 16 

15 

1 

19 

51 

3 

38 

16 

4 

58 

16 15 

16 1 

25 

3 

3 

42 

2 

4 

59 

38 14 

17 

1 

30 

13 

3 

45 

45 

5 

0 

53 13 

j 18 1 

35 

2113 

49 

24 

"5 

2 

3 12 

19 1 

1 

40 

27 

3 

52 

58 

5 

3 

8 11 

120 1 

45 

32 

3 

56 

23 

5 

4 

7 

10 

21 

1 

50 

35 

3 

59 

54 

o 

5 

0 

9 

122 1 

55 

35 

4 

3 

16 

5 

5 

48 

8 

23 

2 

0 

34 

4 

6 

33 

5 

6 

31 

7 

24 

o 

5 

30 

4 

9 

45 

5 

7 

7 

6 

25 

o 

10 

25 

4 

12 

53 

5 

7 

38 

5 

26 

2 

15 

16 

4 

15 

57 

5 

8 

4 

4 

27 

2 

20 

6 

4 

18 

55 

5 

8 

23 

3 

1282 

24 

52 

4 

21 

49 

5 

8 

38 

2 

129 

i 5 

29 

36 4 

24 

38 

5 

8 

46 

1 

30 

! 2 

34 

1C 

4 

27 

23 

5 

8 

49 

0 

S. 



11 



10 

— 9 

S. 

k 

+ 5 


+ 

4 


+ 

3 

s. 


.s. 

+ 

1 o 

1 + 

1 

+ 

2 


: s - 


6 


r» 

/ 


8 

s. 

0 

/ 


f 

// 


n 

0 

00 

04 

24 

7 

38 

30 

1 0 

9,4 

32 

7 

42 29 

20 

184 

40 

7 

46 28 

30 

28 4 

48 

7 

51;27 

40 

37 

4 

55 

7 

55,26 

50 

46| 

5 

3 

7 

5925 

60 

55 

5 

11 

8 

324 

7 

1 

4 

5 

18 

8 

6 23 

8 1 

14 

5 

25 

8 

10 22 

9 1 

23 

5 

32 

8 

1321 

10 1 

325 

40 

8 

16 

20 

11 

1 

41 

5 

47 

8 

20 

19 

12 1 

50 

5 

54 

8 

22 

18 

13 1 

59 6 

0 

8 

25 

17 

142 

86 

j 

7 

8 

28 

16 

15 2 

17 

6 

14 

8 

30 

15 

16 2 

26 6 

20 

8 

33 

14 

17 

2 

346 

26 

8 

35 

13 

18 

2 

43 6 

33 

8 

37 

12 

19 

2 

526 

39 

8 

39 

11 

20 3 

16 

45 

8 

40 

10 

21 3 

96 

51 

8 

42 

9 

22 3 

186 

56 

8 

43 

8 

23 3 

26 7 

2 

8 

44 

7 

24 3 

35 

7 

7 

8 

45 

6 

25 

3 

437 

13 

8 

46 

5 

26 3 

52 7 

18 

8 

47 

4 

27 

4 

0 

7 

23 

8 

48 

3 

28 4 

8 

7 

28 

8 

48 

2 

29 

4 

167 

33 

8 

48 

1 

30 

4 

24 

7 

388 

48 

0 

S. 


-11 


-10 


- 9-S. 

S. 

+ 

5 

-j- 4-j- 3[S. 




























































LUNAR TABLES. 


279 


III. For the Moon's Latitude. 
Argument. 

Arg. I. — D’s Mean Anom. 


s. 

— 0 

— 1 

— 2 

s. 

s. 

+ 6 

+ 7 

+ 8 

s. 

o 

It 

II 

II 

0 

0 

0 

9 

15 

30 

10 

3 

11 

16 

20 

20 

6 

13 

17 

10 

30 

9 

15 

18 

0 

s. 

4-11 

+ 10 

+ 9 

3. 

s. 

— 5 

— 4 

Cl 

— O 

-3. 


IV. For the Moon's Latitude. 
Argument. 

Arg. III. — 3) ’s Mean Anom. 


S. 

— 0 

1 

_ 0 

S. 

s. 

+ 6 

+ T 

+ 8 

S. 

0 

II 

II 

II 

o 

0 

0 

13 

22 

30 

10 

4 

16 

24 

20 

20 

9 

19 

25 

10 

30 

13 

22 

25 

0 

S. 

+ 11 

+ 10 + 9 

.3. 

s. 

_ 5 

- 4 

- 3 

3. 


V. For the Moon's Latitude. 
Argument. 

Arg. IV. — 3) ’s Mean Anom. 


S. 

4- o 

+ 1 

+ 2|S. 

S. 

— 6 

— 7 

— 8 

S. 

o 

// 

II 

il 

0 

0 

0 

8 

14 

30 

10 

3 

10 

15 

20 

20 

5 

12 

16 

10 

30 

8 

14 

16 

0 

s. 

—11 

—10 

— 9 

s. 

s. 

+ 5 

+ 4 

+ 3 

s. 


FOR THE MOON’S EQUATORIAL PARALLAX. 


I. Argument V. of Longitude, viz. 2 J) 
from © — jj's Mean Anomaly. 


II. Argument XI. of Longitude, viz. correct 
Anomaly of ]). 


s. 

— 0 

— 1 

— 2 

+ 3+ 4|+ 5 

S. 

o 

II 

II 

II 

II 

ll 

II 

o 

0 

37 

32 

19 

0 

18 

32 

30 

5 

37 

30 

16 

3 

21 

34 

25 

10 

36 

28 

13 

6 

24 

35 

20 

15 

36 

26 

10 

9 

26 

36 

15 

20 

35 

24 

7 

13 

29 

37 

10 

25 

34 

21 

3 

16 

31 

37 

5 

30 

32 

19 

0 

18 

32 

38 

0 

S. 

—11 

—10 

— 9 

+ 8,+ 7 

+ 6 

s. 


+ 0 


13 


+ 1 


+ 2 


54 33 55 32 


14 54 41 i55 46 


10 54 
1554 18 
20 54 22 


15 54 49[55 59 

54 58 56 14 

55 9l56 29 


25;54 27 55 20|56 45 
30 54 33 55 32 57 1 


S. 


+ 11+10 + 9 


+ 3 


57 

57 

57 

57 

58 


1 

17 

34 

51 

7 


+ 4 ; + 5 


58 39 59 59 

58 55 60 8 

59 10 60 15 
59 24 60 21 
59 36 60 25 


58 24 59 48 60 28 


58 39 59 59 


+ 8 


+ 7 


60 29 


+ 6 


S. 












































































280 


APPENDIX TO THE ASTRONOMY 




III. Argument XII. of Longitude, viz. 5’s 
equated Long. — ©*s true Long. 


s. 

+ 0 

± 1 

— 2 

— 3 

±4+5 

S. 

0 

it 

// 

n 

// 

" 

" 

0 

0 

25 

12 

14 

26 

13 

14 

30 

5 

25 

8 

n 

25 

O 

O 

18 

25 

10 

24 

+ 3 

20 

24 

— 4 

21 

20 

15 

22 

— 1 

23 

22 

+ 0 

24 

15 

20 

19 

5 

24 

19 

5 

25 

10 

25 

16 

10 

25 

16 

10 

27 

5 

30 

12 

14 

26 

13 

14 

27 

0 

S. 

4-11 

±10 

— 9 

— 8 

=F ^ 

+ 6 

s. 


TABLE/or finding the Moon’s Diameter. 


Argument. / Equatorial Parallax. 


Equat. Far. 

i) ’s Diam. 

Equat. Far. 

j ’s Diam. 

f 

// 

f 

It 

/ 

" 

/ 

It 

54 

0 

29 

26 

58 

0 

31 

37 

54 

10 

29 

31 

58 

10 

31 

42 

54 

20 

29 

37 

58 

20 

31 

47 

54 

30 

29 

42 

58 

30 

31 

53 

54 

40 

29 

48 

58 

40 

31 

58 

54 

50 

29 

53 

58 

50 

32 

4 

55 

0 

29 

58 

59 

0 

32 

9 

55 

10 

30 

4 

59 

10 

32 

15 

55 

20 

30 

9 

59 

20 

32 

20 

55 

30 

30 

15 

59 

30 

32 

26 

55 

40 

30 

20 

59 

40 

32 

31 

55 

50 

30 

26 

59 

50 

32 

36 

56 

0 

30 

31 

60 

0 

32 

42 

56 

10 

30 

37 

60 

10 

32 

47 

56 

20 

30 

42 

60 

20 

32 

53 

56 

30 

30 

47 

60 

30 

32 

58 

56 

40 

30 

53 

60 

40 

33 

4 

56 

50 

30 

58 

60 

50 

33 

9 

57 

0 

31 

4 

61 

0 

33 

15 

57 

10 

31 

9 

61 

10 

33 

20 

57 

20 

31 

15 

61 

20 

33 

26 

57 

30 

31 

20 

61 

30 

33 

31 

57 

40 

31 

26 

61 

40 

33 

36 

57 

50 

31 

31 

61 

50 

33 

42 

58 

0 

31 

37 

62 

0 

33 

47 ( 


TABLE/or the reduction of lat. and 
tlor. Par. for Ellipticity. g-L 




Reduct. 

Red. 

Hor.Par- 

o 

Lat. 

of Lat. 

Horizontal Par. 

0 

3 

nj 

0 

u 



53' 

57' 

61' 






V 

O 

It 

// 

n 

ff 

*-> 

0 

0. 0.0 

0.0 

00 

0.0 

bO 

2 

0.47.9 

0.0 

0.0 

0.0 

c 

'<5 

4 

1.35.5 

0.1 

0.1 

0.1 

0) 

c 

6 

2.22.7 

0.1 

01 

0.1 

<*-4 

8 

3. 9.2 

0.2 

0 2 

0.2 

0) 

10 

3.54.8 

0.3 

0.3 

0.4 

G 

cO 

<D 

12 

4.39.3 

0.5 

0.5 

0.5 


14 

5.22.4 

0.6 

0.7 

0.7 

G 

16 

6. 3.9 

0.8 

0.9 

0.9 

-O 

18 

6.43.7 

1.0 

1.1 

1.2 

a 

<D 

20 

7.21.5 

1.2 

1.3 

1.4 

DG 

*-> 

22 

7.57 2 

1.5 

1.6 

1.7 

O 

24 

8.30.7 

1.8 

1.9 

20 

K 

C3 

26 

9. 1.6 

2.0 

2.2 

2.3 


28 

9.29.9 

2.3 

2.5 

2.7 

a 

cu 

0 

30 

9.55.4 

2.7 

2.9 

3.1 

32 

10.18.1 

3.0 

3.2 

3.4 

cS 

3 

34 

10,37.8 

3.3 

3.6 

3.8 


36 

10.54.3 

3.7 

3.9 

4.2 

Ja 

38 

11. 77 

4.0 

4.3 

4.6 

•g-s 

« 3 

40 

11.17.8 

4.4 

4.7 

50 

c IS 
s - 

42 

11.24.7 

4 7 

5.1 

5.5 

J.S 

44 

11.28.2 

5 1 

5.5 

5.9 

O 

O 

46 

11.28.4 

5.5 

5.9 

6.3 

C G 

48 

11.25.1 

5.9 

6.3 

6.7 


50 

1 1.18.6 

6.2 

6.7 

7.2 







~ "a 






<D t* 

52 

11. 8.8 

6.6 

7.1 

7.6 

= 5 

54 

10.55.6 

6.9 

7.5 

8.0 

CO 

<D •- 

56 

10.39.3 

7.3 

7.8 

8.4 

« B 

4 s 

58 

10.19.9 

7.6 

8.2 

8.8 

« 0 

60 

9.57.4 

7.9 

8.5 

9.1 

<■«-. ^5 
© G 

62 

9.32.0 

8.3 

8.9 

9.5 

<u 0 
'a g 

64 

9. 3.8 

8.6 

9.2 

9.9 

-2 ^ 

66 

8.32;9 

8.8 

9.5 

10.2 


68 

7.59.6 

9.1 

9.8 

10.5 

^.s 
« 2 

70 

7.23.8 

9.4 

10.1 

10.8 

CD ^ 

3 2 

CO s 

H 

k CO 

0) 

72 

6.45.9 

9.6 

10.3 

11.0 

74 

76 

6. 6.0 
5.24.3 

9.8 

10.0 

10 5 

10.7 

11.3 

11.5 

rG ^ 

78 

4.41.0 

10.1 

10.9 

11.7 

0 .5 

80 

3.56 3 

10.3 

11.1 

11.8 

= '2 






l! 

82 

3.10 4 

10.4 

11.2 

12.0 


84 

2.23.7 

10.5 

11.3 

12.1 

CC ‘ —' 

CO 

86 

1.36.2 

10.5 

11.3 

12.1 

_a cc 

H ^ 

88 

0.48.2 

10.6 

11.4 

12.2 

0 1 

90 

0. O.U 

10.6 

11.4 

12.2 


i 































































LUNAR TABLES 


281 


TABLES FOR FINDING THE MOON’S HOURLY MOTION IN LONGITUDE 

AND LATITUDE. 


I. For the Moon's Hourly Motion in Long. 

Arg. V. of Long. viz. 2 3 from © — j) ’s 
Mean Anomaly. 


s. 

— 0 

— 1 

— 2 

+ 

+ 4 

+ 5 

s. 

0 

II 

// 

It 

It 

II 

It 

0 

0 

41 

36 

21 

0 

21 

37 

30 

5 

11 

34 

18 

3 

24 

38 

25 

10 

41 

32 

15 

7 

27 

40 

20 

15 

40 

30 

11 

10 

30 

41 

15 

20 

39 

27 

8 

14 

32 

42 

10 

25 

38 

24 

4 

17 

35 

42 

5 

30 

36 

21 

0 

21 

37 

43 

0 

S. 

— 11 

^10 

— 9 

+ 3 

4- 7 

+ 6 

S. 


* 

II. For the Moon's Hourly Motion in Long. 

Arg. Xl. of Long, viz the correct Anom¬ 
aly of the 2 • 


s. 

+ 0 

+ 

1 

+ 2 

+ 

3 

+ 

4 

+ 

5 

s. 

o 

' 

>1 

< 

•l 

1 

It 

t 

// 

1 

il 

/ 

// 

o 

0 

29 

35 

29 

bl 

31 

2 

32 

42 

34 

36 

36 

10 

30 

5 

29 

35 

30 

5 

31 

17 

33 

o 

34 

54 

36 

21 

25 

10 

29 

37 

30 

15 

31 

32 

33 

19 

35 

11 

36 

30 

20 

15 

29 

40 

30 

25 

31 

49 

33 

39 

35 

28 

36 

38 

15 

20 

29 

45 

30 

36 

32 

6 

33 

58 

35 

43 

36 

43 

10 

25 

29 

50 

30 

49 

32 

23 

34 

17 

35 

57 

36 

46 

5 

30 

29 

57 

31 

2 

32 

42 

34 

36 

36 

10 

36 

48 

0 

s. 

+ 

11 

+ 

10 

+ 9 

+ 

8 

+ 

7 

+ 

6 

S. 


III. For the Moon's Hourly Motion in Long. 

Arg. XII. of Long. viz. D’s equat. 
Long.—Q’s true Long. 


I. For the Moon's Hourly II. For the Moon's Hourly 
Motion in Lat. Motion in Lat. 

Argument I. of Lat. Argument II. of Lat. 


S. 

+ 0 

±z 1 

— 2 

— 3 

=F 4 

+ & 

S. 

0 

II 

II 

H 

•! 

II 

II 

o 

0 

40 

19 

21 

40 

20 

21 

30 

5 

39 

13 

26 

39 

13 

27 

25 

10 

37 

+ 6 

31 

37 

— 7 

32 

20 

15 

34 

— 1 

35 

34 

+ o 

36 

15 

20 

30 

8 

38 

30 

7 

39 

10 

25 

25 

15 

39 

25 

14 

41 

5 

30 

19 

31 

40 

20 

21 

42 

0 

S. 

+M. 

i=10 

— 9 

— 8 

=F 7 

+ 6 

S. 


S. 

4- 0 

+ 1 

+ 2 

S. 

s. 

— 6 

— 7 

— 8 

S. 

0 

t II 

/ II 

1 II 

o 

0 

2 58 

2 34 

1 29 

30 

5 

2 57 

2 26 

1 15 

25 

10 

2 55 

2 16 

1 1 

20 

15 

2 52 

2 6 

0 46 

15 

20 

2 47 

1 54 

0 31 

10 

25 

2 41 

1 42 

0 15 

5 

30 

2 34 

1 29 

0 0 

0 

S. 

+ 11 

+ 10 

+ 9 

S. 

S. 

— 5 

— 4 

— 3 

S. 


s. 

f- o 

+ 1 

+ 2 

S. 

s. 

— 6 

— 7 

— 8 


o 

f 

// 

II 

0 

0 

4 

4 

2 

30 

5 

4 

3 

2 

25 

10 

4 

3 

1 

20 

15 

4 

3 

l 

15 

20 

4 

3 

1 

10 

25 

4 

2 

0 

5 

30 

4 

2 

0 

0 

S. 

+ 11 

+ 10 

+ 9 

s. 

s. 

— 5 

— 4 

— 3 

S7 


t 


39£ 
















































































282 


APPENDIX TO THE ASTRONOMY 


Angle of the visible Path of the Moon with the Ecliptic in 
Eclipses. 

Argument. Long. D in her orbit — Long. ft. 


Tables of the Mean Motions of the Moon from 
the Sun. 


0 Signs 6. 


Horary Motion of the Moon from the Sun. 



t 

/ 

/ 

/ 


27 

28 

29 

30 

0 

0 / 

0 / 

o / 

O t 

0 

5 47 

5 46 

5 45 

5 44 

3 

5 46 

5 45 

5 44 

5 43 

6 

5 45 

5 44 

5 43 

5 42 

9 

5 42 

5 41 

5 40 

5 39 

12 

5 39 

5 38 

5 37 

5 36 

15 

5 35 

5 34 

5 33 

5 32 


1 


31 


5 43 
5 42 
5 41 


5 39 
5 35 
5 31 


32 


5 42 
5 42 
5 40 


5 38 
5 35 
5 31 


33 


5 41 
5 41 
5 39 


5 37 
5 34 
5 30 


34 


5 41 
5 40 
5 39 


5 37 
5 33 
5 29 


35 


36 


5 40 5 39 
5 40 5 39 
5 38 5 38 


5 36 
5 33 
5 29 


5 35 
5 32 
5 28 


11 Sisrns 5. 


A.D.[S. 0 ' " 

1761: 9 20 47 17 

1781 2 4 12 47 

1791 10 4 49 47 

B 1792 2 26 38 39 

1793 7 6 16 4 

** | 

1794 

1795 
B 1796 

1797 

1798 

11 15 53 29 
3 25 30 54 
8 17 19 45 
0 26 57 10 
5 6 34 35 

1799 

1800 
1801 
1802 
1803 

9 16 12 0 

1 25 49 25 

6 5 26 49 

10 15 4 15 

2 24 41 39 

B 1804 

1805 

1806 
1807 

B 1808 

7 16 30 31 

11 26 7 56 

4 5 45 20 

8 15 22 45 

1 7 11 37 

1809 

1810 
1811 

B 1812 
1813 

5 16 49 3 
9 26 26 28 
2 6 3 54 

6 27 52 47 
11 7 30 12 

1814 

1815 
B 1816 

1817 

1818 

3 17 7 38 
7 26 45 3 
0 18 33 56 

4 28 11 22 
9 7 48 47 

1819 
B 1820 

1821 

1 17 26 13 
6 9 15 6 

10 18 52 32 


Years Complete. 


S. ° • " 

1 

4 9 37 24 

2 

8 19 14 49 

3 

0 28 52 14 

B 4 

5 20 41 6 

5 

10 0 18 30 

6 

2 9 55 54 

7 

6 19 33 19 

B 8 

11 11 22 10 

9 

3 20 59 36 

10 

8 0 37 0 

11 

0 10 14 24 

B 12 

5 2 3 16 

’ ,13 

9 11 40 40 

14 

1 21 18 5 

15 

6 0 55 30 

B 16 

10 22 44 21 

17 

3 2 21 46 

18 

7 11 59 10 

19 

11 21 36 35 

B 20 

4 13 25 26 


Months. 


S. ° ' " 

January 

0 0 0 0 

F ebruary 

0 17 54 48 

March 

11 29 15 16 

April 

0 17 10 3 

May 

0 22 53 23 

June 

1 10 48 11 

July 

1 16 31 22 

August 

2 4 26 20 

Septem. 

2 22 21 9 

October 

2 28 4 29 

Novem. 

Decern.^ 

3 15 5P 16 
3 21 42 37 


In months after February of bissextile years subtract 
one day from the time found by the Tables. 














































LUNAR TABLES 


283 


TABLES of the Mean Motions of the Moon TABLE I. Of Mean New Moons, #c. in March, from 1791 
from the Sun. * to 1821. 


Davs. 



S. 

0 

' 


1 

0 

12 

11 

27 

2 

0 

24 

22 

53 

3 

1 

6 

34 

20 

4 

1 

18 

45 

47 

5 

2 

0 

57 

13 

6 

2 

13 

8 

40 

7 

2 

£5 

20 

7 

8 

3 

7 

31 

33 

9 

3 

19 

43 

0 

10 

4 

1 

54 

27 

11 

4 

14 

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IV. The Annual or First Equation of the Mean to the trice Syzygy • V. Equation of the Moon’s Mean Anomaly 

i 

Argument. Sun’s Mean Anomaly. Argument. Sun’s Mean Anomaly. 


285 


LUNAR TABLES. 



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286 


APPENDIX TO THE ASTRONOMY. 


VI. The Second Equation of the Mean to the True 
Syzygy- 


VII. 


The third Equation VIII. The fourth Equation 


of the Mean to the 
True Syzygy 


of the Mean to the 
True Syzygy. 


Argument. Moon’s equated Anomaly. 


Arg. ©’s Mean Anotn. Argument, ©’s Mean 
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102 

304 

18 

28 


2 0 

7 

1 

24 

l 

20 

28 

3 

0 

152 

34 4 

21 

27 


3 0 

10 

1 

25 

1 

18 

27 

4 

0 

20 2 

38)4 

24 

26 


4 

0 

13 

1 

26 

1 

16 

26 

5 

0 

252 

424 

27 

25 


5 

0 

16 

1 

,7 

1 

14 

25 

6 

0 

30 2 

46j4 

30 

24 


6 0 

20 

1 

28 

i 

12 

24 

7 

0 35 2 

504 

32 

23 


7 

0 

23 

1 

29 

1 

10 

23 

8 

0 

40 2 

544 

4 

22 


8 

0 

26 

1 

30 

l 

8 

22 

9 

0 

452 

58 

4 

36 

21 


9 

0 

29 

1 

31 

l 

6 

21 

10 

0 

503 

2 

4 

38 

20 


10 

0 

32 

1 

32 

1 

3 

20 

11 

0 

553 

6 

4 

40 

19 


11 

0 

35 

1 

33 

l 

0 

19 

12 

I 1 

03 

.! 

10 

4 

42 

18 


12 

0 

38 

1 

33 

0 

57 

18 

13 

1 

5l3 

14 

4 

44 

17 


13 

0 

41 

1 

34 

0 

54 

17 

14 

1 

103 

18 

4 

46 

16 


14 

0 

441 

34 

0 

51 

16 

15 

1 

153 

22 

4 

48 

15 


15 

0 

47 1 

34 

0 

49 

15 

16 

1 

20 3 

6 

4 

50 

14 


16 

0 

50 1 

34 

0 

45 

14 

17 

1 

25 

3 

30 

4 

51 

13 


17 

0 

52 1 

34!0 41 

13 

18 

1 

30 

3 

34 

4 

52 

12 


18 

0 

54 1 

310 

37 

12 

19 

1 

35 

3 

38 4 

53 

il 


19 

0 

57 1 

33 

0 

34 

11 

20 

1 

40 

3 

42 4 

54 

10 


20 

1 

0 1 

330 

31 

10 

21 

1 

45 

3 

45 4 

55 

9 


21 

1 

21 

320 

28 

9 

22 

1 

49 

3 

48 4 

56 

8 

122 

1 

5,1 31 

0 

25 

8 

23 

1 

52 

3 

51 4 

57 

7 

23 

1 

8 1 

300 

22 

7 

24 

1 

56 

3 

544 

57 

6 

24 

1 

10 1 

29j0 

19 

6 

25 

2 

0 

3 

57 4 

57 

5 

25 

1 

12 1 

28 

0 

16 l 

5 

26 

2 

4 

4 

04 

58 

4 

26 

1' 

14 1 

27 

0 

13 ! 

4 

27 

2 

9 

4 

34 

58 

3 

127 

1 

16,1 

26,0 

10 

3 

23 

2 

13 

4 

6 4 

58 

2 

28 

1 

18 1 

25, 

0 

6 

2 

29 

2 

18 

4 

94 

58 

1 

29 

1 

20‘1 

24 

0 

3 

1 

30: 

2 

22 

4 

12 4 58 

0 


30 ( 

1 

221 

22 

0 

0 

0 


S.- 

-5 


.4 


-3 S. 




S.- 

-5 


- 4 


-3 S. 


1 

S.+l! 

+ i6]+9 S. 

’ 

V 


s.- 

1 1 


-io| 


-9 S. 




































































LUNAR TABLES 


287 


TABLE of Logistical Logarithms. 



0 

t 

2 

3 

4 

5 

6 . 7 


6 

1 

2 

3 

4 

5 

6 1 

7 


0 

60 

120 

180 

240 

300 

360 420 


0 

60 

120 

180 

240 

300 

360 

420 

0 

ooooo 

17782 

14771 

13010 

11761 

10792 

10000 9331 

30 

20792 

16021 

13802 

12341 

11249 

10378 9652 9031 

1 

85563 17710 

14735 

12986 

11743 

10777 

9988 9320 

31 

20619 

15973 

13773 

! 2320 

11233 

10365 9641 9021 

2 

32553 17639 

14699 

12962 

11725 

10763 

9976 9310 

32 

20512 

15925 

13745 

12300 

1 1217 

10352 9630 9012 

3 

3079217570 

14664 

12939 

11707 

10749 

9964 9300 

33 

20378 

15878 

13716 

12279 

11201 

1033919619900, 

4 

29542 17 501 

:4629 

12915 

11683 

10734 

9952 9289 

34 

20248 

15832 

13688 

12259 

11186 

10326 

9608 8992 

5 

2857317434 

14594 12891 

11671 

10720 

9940 9279 

35 

20 [22 

15736 

13660 

12239 

11170 

10313 

9597 8983 

6 

27782 17368 

1455912868 

11654 

10706 

9928 9269 

36 

20000 

15740 

13632 

12218 

11154 

10300 

95868973 

7 

2711217302 

14525 

12845 

11636 

10692 

9916 9259 

37 

19881 

15695 

13604 

12198 

11138 

10287 

9575 8964 

8 

26532 

17238 

144911 

12821 

11619 

10678 

9905 9249 

38 

19765 

15651 

13576 

12178 

11123 

10274 

9564 8954 

9 

26021 

17175 

144571 

12798 

11601 

10663 

9893 9238 

39 

19652 

15607 

13549 

12159 

11107 

10261 

9553 8945 

10 

2o563 

17112 

14424 

12775 

11584 

10649 

9881 9228 

40 

19542 

15563 

13522 

12139 

11091 

10248 

9542 8935 

- 

11 

25149 

17050 

14390 

12753 

11566 

10635 

9869 9218 

| 

41 

19435 

15520 

13495 

12119 

11076 

10235 

95328926 

12 

24771 

16990 

14357 

12730 

11549 

10621 

9858 9208 

42 

19331 

15477 

13468 

12099 

11061 

10223 

9521,8917 

I O 

24424 

16930 

14325 

12707 

11532 

10608 

9846 9198 

43 

19228 

15435 

13441 

12080 

11045 

10210 

95108907 

1 1 

24102 

16871 

14292 

12685 

11515 

10594 

98349188 

44 

19128 

15393 

13415 

12061 

11030 

10197 

9499 8898 

15 

23802 

16812 

14260 

12663 

11498 

10580 

9823 9178 

l 

45 

i 

19031 

15351 

13388 

12041 

11015 

10185 

9488 8888 

16 

23522 

16755 

14228 

12640 

11481 

10566 

9811 9168 

46 

18935 

15310 

13362 

12022 

10999 

10172 

94788879 

17 

23259 

16698 

14196 

12618 

11464 

10552 

9800 9158 

47 

18842 

15269 

13336 

12003 

10984 

10160 

9467.8870 

18 

23010 16642 

14165 

12596 

11 147 

10539 

9788 9148 

48 

18751 

15229 

13310 

1'984 

10969 

10147 

9456 ; 8861 

19 

22775 16587 

14133 12574 

11430 

10525 

9777 9138 

49 

18661 

15139 

13284 

11965 

10954 

10135 

9446 8851 

20 

22553 16532 

14102 

12553 

11 113 

10512 

9765,9128 

50 

18573 

15149 

13259 

11946 

10939 

10122 

94358842 

21 

22341 16478 

14071 

12531 

11397 

10498 

9754 9 119 

51 

18487 

15110 13233 

11927 

10924 

10110 

9425 8833 

22'22139 16125 

14040 

1 ,'510 

11380 

10484 

9742 9109 

52 

18403 

15071 

13208 

11908 

10909 

10098 

9414|8824 

23 

z 1946 16372 

14010 

12488 

11363 

10471 

9731 9099 

53 

18320 

15032 

113183 

11889 

10894 

f0085 

94048814 

24 

21761 16320 

13979 

12467 

11347 

10458 

9720 9089 

54! 18239 

14994 

.13158 

11871 

10880 

10073 

93938805 

2F 

>2158416269 

13949 

12445 

11331 

10444 

9708 9079 

55 

18159 

14956 

13133 

11852 

10865 

10061 

93838796 

21 

21411 

516218 

13919 

1242'! 

11314 

10431 

9697 907C 

56 

18031 

14918 

13108 

11834 

10850 

10049 

9372 

8787 

2 " 

21241 

>16168 

13896 

12408 

11298 

10418 

9686 9066 

51 

1800-1 

14881 

13083 

11816 

10835 

10036 

9362 

8778 

21 

5 21091 

16118 

13866 

12382 

11282 

10404 

9675 905C 

56 

1792S 

14844 

1305S 

11791 

10821 

10024 

9351 

8769 

•vC 

209391606? 

13831 

12362 

11266 

10391 

9664 9w41 

5£ 

17855 

14806 

13034 

1177E 

10806 

10012 

9341 

8760 

( 3C 

>2079216021 

13802 

12341 

11246 

10378 

9652 9031 

<66 

) 1778? 

14771 

13016 

>11761 

10792 10000 

9331 

8752 






































































































288 


APPENDIX TO THE ASTRONOMY. 


TABLE of Logistical Logarithms. 



8 

9 

10 

11 

12 

13 

14 

15 

16 


8 

9 

10 

11 

12 

13 

14 

15 

16 ' 


480 

540 

600 

660 

720 

780 

840 

900 

960 


180 

540 

600 

660 

720 

780 

840 

900 

960 

08751 

8239 

7782 

7368 

6990 

6642 6320 

6021 

5740 

30 

8487 

80.4 

7573 

717' 

6812 

6478 

6168 

5878 

5607 

1 

8742 

8231 

7774 

7361 

6984 

6637 6315 

6016 

5736 

31 

8479 

7997 

7563 

7168 

6807 

6473 

6163 

5874 

5602 

2 

8733 

8223 

7767 

7354 

6978 

663) 6310 

6011 

3731 

32 

8470 

7989 

7556 

7162 

6801 

6466 

6158 

5879 

5598 

3 

8724 

8215 

7760 

7348 

6972 

6625 6305 

6006 

5727 

33 

8462 

7981 

7549 

7156 

6795 

6462 

6153 

5864 

5594 

4 

8715 

8207 

7753 

7341 

6966 

6620 6300 

6001 

5722 

34 

8453 

7974 

7542 

7149 

6789 

6457 

6148 

5860 

5589 

5 

8706 

8199 

7745 

7335 

6960 

6614 

6294 

6997 

5718 

35 

8445 

7966 

7535 

7143 

6784 

6451 

6143 

5855 

5585 

6 

8697 

8191 

7738 

7328 

6954 

6609 

6289 

5992 

5713 

36 

8437 

7959 

7521 

7 37 

6778 

6446 

6138 

5850 

5580 

7 

8688 

8183 

7731 

7322 6948 6603 

6284 

5987 

5709 

37 

8428 

7951 

7522 

7131 

6772 

6441 

6133 

5846 

5576 

8 8679 

8175 

7724 

73156942 

6598 

6279 

5982 

5704 

38 

8420 

7944 

7515 

7124 

6766 

6435 

6128 

6841 

5572 

9,8670 

8167 

7717 

7309 6336 

6592 

6274 

5977 

5700 

39 

8411 

7936 

7508 

7118 

6761 

6430 

6123 

5836 

5567 

10 

8661 

8159 

7710 

7302 6930 

6587 

6269 

5973 

5695 

40 

8403 

7929 

7501 

7112 

6755 

6425 

6118 

5832 

5563 

11 

8652 

8152 

7703 

7296 6924 

6581 

6264 

5968 

5691 

41 

8395 

7921 

7494 

7101 

6749 

6420 

6113 

5827 

5559 

12 

8643 

8144 

7696 

7289 6918 

6576 

6259 

5963 

5686 

42 

8336 

7914|7488 

7100 

6743 

6414 

6108 

5823 

5554 

13 

8635 

8136 

7688 

7283 6912 

6570 

6254 

5958 

5682 

43 

8378 

7906 7481 

7093 

6738 

6409 

6103 

5818 

5550 

14 

8626 

8128 

768 

7276 6906 

6565 

6248 

5954 

5677 

44 

8370 

7899 7474 

7087 

6732 

6404 

6099 

5813 

5546 

15 

8617 

8120 

7674 

7270 

6900 

6559 

6243 

5949 

5673 

45 

8361 

7891 7467 

7081 

6726 

6398 

6094 

5809 

5541 

16 8608 

8112 

7667 

7264 

6894 

6554 

6238 

5944 

5669 

46 

8353 

7884 7461 

7075 

6721 

6393 

6089 

5804 

5537 

1 7 8589 

8101 

7660 

7257 

6888 

6548 6233 

5939 

5664 

47 

8345 

7877 

7454 

7069 

6715 

6388 

6084 

5800 

5533 

188591 

8097 

7653 

7251 

6882 

6543 6228 

5935 

5660 

48 

8337 

7869 7447 

7063 

6709 

638.3 

6079 

5795 

5528 

19 8582 

8089 

7646 

7244 

6877 

6538 6223 

5930 

5655 

49 

8328 7862 7441 

7057 

6704 

6377 

6074 

5790 

5524 

20 8573 

8081 

7639 

7238 

6871 

6532 6218 

5925 

5651 

50 

8320 7855 7434 

7050 

6698 

6372 

6069 

5786 

5520 

21,8565 

80737632 

7232 

6865 

6527 6213 

5920 

5640 

51 

8312 7847 7427 

7044 

669 

6367 

6064 

781 

5516 

22 8556 

806617625 

7225 

6859 

6521 6208 5916 

5642 

52 

8304 

7840 

7421 

7038 

6687 

6362 

■r '51 

5771 

5511 

23 

8547 

8058,7618 

7219 

6853 

6516 6203 5911 

5637 

53 

3296 

7832 

7414 

7032 

6681 

6357 

6058 

772 

5507 

24 

8539 

8050 7611 

7212 

6817 

6510 6198 5906 

5633 

54 

8288 

7825 

7407 

7026 

6676 

6351 

60'0 

5768 

5503 

25 

8530 

80437604 

7206 

6841 

6505 

6193 5902 

5629 

55 

8279 

7818 

1401 

7020 

6670 

6346 

6045 

5763 

5498 

26 

8522 

80357597 

7200 

6836 

6500 

6188 

5897 

5624 

56 

8271 

7811 

7394 

7014 

6664 

6341 

6040 

758 

5494 

27 

8513 

802717590 7193 

6830 

6494 

6183 

5892 

5620 

57 

8263 

7803 

7387 

7008 

6659 

633» 

6035 

5754 

5490 

28 

8504 

8020 7583 7187 

6824 

6489 

6178 

5888 

5615 

58 

8255 

7796 

7381 

7002 

6653 

6331 

6030 

5749 

5486 

29 

8496 

8012 7577 7181 

6818 

6484 

6173 

5883 

5611 

59 

8247 

7789 

7374 

6996 

6648 

6325 

6025 

5745 

5481 

30 8487 

8004(7570 7175 

6812 

6478 

6168 

5878 

5607 

60 

8239 

7782 

7368 

6990 

6642 

6320 

>021 

5740 

5477 























































































LUNAR TABLES 


289 


TABLE of Logistical Logarithms. 



17 

18 

19 

20 

21 

22 

23 

24 

25 || 

17 

18 

19 

__ 

O 

Of 

22 

23 

24 

25 


1020 

1080 

1140 

1200 

1260 

1320 

1380 1440 

1500 


1020 

1080 

1140 

12001260 

1320 

1380 

1440 

1500 

0 

5477 

5229 

4994 

4771 

4559 

4357 

4164 

3979 

3802 

30 

5351 

5110 

4881 

4664 4457 

4260 

4071 

3890 

3716 

1 

5473 

5225 

4990 

4768 

4556 

4354 

4161 

3976 

3799 

31 

5347 

5106 

4877 

4660 4454 

4256 

4068 

3887 

3713 

2 

5469 

5221 

4986 

4764 

4552 

4351 

4158 

3973 

3796 

32 

5343 

5102 

4874 

4657 4450 

4253 

4065 

3884 

3710 

3 

5464 

5217 

4983 

4760 

4549 

4347 

4155 

3970 

3793 

33 

5339 

5098 

4870 

4653 4447 

4250 

4062 

3881 

3708 

4 

5460 

5213 

4979 

4757 

4546 

4344 

4152 

3967 

3791 

34 

5335 

5094 

4866 

4650 4444 

4247 

4059 

3878 

3705 

5 

5456 

5209 

4975 

4753 

4542 

4341 

4149 

3964 

3788 

35 

5331 

5090 

4863 

4646 4440 

4244 

4055 

3875 

3702 

G 

5452 

5205 

4971 

4750 

4539 

4338 

4145 

3961 

3785 

36 

5326 

5086 

4859 

46434437 

4240 

4052 

3872 

3699 

7 

5447 

5201 

4967 

4746 

4535 

4334 

4142 

3958 

3782 

37 

5322 

5082 

4855 

4639 4434 

4237 

4049 

3869 

3696 

8 

5443 

5197 

4964 

4742 

4532 

4331 

4139 

3955 

3779 

38 

5318 

5079 

4852 

4636 4430 

4234 

4046 

3866 

3693 

9 

5439 

5193 

4960 

4739 

4528 

4328 

4136 

3952 

3776 

39 

5314 

5075 

4848 

46324427 

4231 

4043 

3863 

3691 

10 

5435 

5189 

4956 

4735 

4525 

4325 

4133 

3949 

3773 

40 

5310 

5071 

4844 

4629 4424 

4228 

4040 

3860 

3688 

11 

5430 

5185 

4952 

4732 

4522 

4321 

4130 

3946 

3770 

41 

5306 

5067 

4841 

4625 4420 

4224 

4037 

3857 

3685 

12 

5426 

5181 

4949 

4728 

4518 

4318 

4127 

3943 

3760 

42 

5302 

5063 

4837 

46224417 

4221 

4034 

3855 

3682 

13 

5422 

5177 

4945 

4724 

4515 

4315 

4124 

3940 

3765 

43 

5298 

5Q59 

4833 

4618 4414 

4218 

4031 

3852 

3679 

14 

5418 

5173 

4941 

4721 

4511 

4311 

4120 

3937 

3762 

44 

5294 

5055 

4830 

46154410 

4215 

4028 

3849 

3677 

15 

5414 

5169 

4937 

4717 

4508 

4308 

4117 

3934 

3759 

45 

5290 

5051 

4826 

4611 4407 

4212 

4025 

3846,3674 

16 

5409 

5165 

4933 

4714 

4505 

4305 

4114 

3931 

3756 

46 

5285 

5048 

4822 

4608 4404 

4209 

4022 

3843 

3671 

17 

5405 

5161 

4930 

4710 

4501 

4302 

4111 

3928 

3753 

|47 

5281 

5044 

4819 

46044400 

4205 

4019 

3840 

3668 

18 

5401 

5157 

4926 

4707 

4498 

4298 

4108 

3925 

3750 

48 

5277 

5040 

4815 

460114397 

4202 

4016 

3837 

3665 

19 

5397 

5153 

4922 

4703 

4494 

4295 

4105 

3922 

3747 

49 

5273 

5036 

4811 

459714394 

4199 

4013 

3834 

3663 

20 

5393 

5149 

4918 

4699 

4491 

4292 

4102 

3919 

3745 

60 

5269 

5032 

4808 

4594 4390 

4196 

4010 

3831 

3660 

21 

5389 

5145 

4915 

4696 

4488 

4289 

4099 

3917 

3742 

51 

5265 

5028 

4804 

4590 4387 

4193 

4007 

3828 

3657 

22 

5384 

5141 

4911 

4692 

4484 

4285 

4096 

3914 

3739 

52 

5261 

5025 

4800 

4587 4384 

4189 

4004 

3825 

3654 

23 

5380 

5137 

4907 

4689 

4481 

4282 

4092 

3911 

3736 

'53 

5257 

5021 

4797 

45844380 

4186 

4001 

3822 

3651 

24 

5376 

5133 

4903 

4685 

4477 

4279 

4089 

3908 

3733 

54 

5253 

5017 

4793 

45804377 

4183 

3998 

3820 

3649 

25 

5372 

5129 

4900 

4682 4474 

4276 

4086 

3905 

3730 

55 

5249 

5013 

4789 

4577-4374 

1 

4180 

3995 

3817 

3646 

26 

5368 

5125 

4896 

4678 4471 

4273 

4083I3902 

3727 

56 

5245 

3009 

4786 

4573 4370 

4177 

3991 

3814 

3643 

27 

5364:5122 

4892 

4675 

4467 

4269 

40803899 

3725 

57 

5241 

5005 

4782 

4570,4367 

4174 

3988 

3811 

3640 

28 

5354 5118 

4889 

4671 

4464 

4266 

4077(3896 

3722 

158 

5237 

5002 

4778 

4566 4364 

4171 

3985 

3808 

3637 

29 

5355 5114 

4885 

4668 

4460 

4263 

40743893 

3719 

59 

5233 

4998 

4775 

4563 l 4361 

4167 

3982 

3805,3635 

30 

5351 

5110 

4881 

4664 

4457 

4260 

4071 >3890 

3716 

'60 

5229 

4994 

4771 

4559k357 

4164 

3979 

380213632 


4U 














































































290 


APPENDIX TO THE ASTRONOMY. 


TABLE of Logistical Logarithms. 



26 

27 

28 

29 

30 

31 

32 

33 

34 


26 

27 

38 

29 

30 

1.31 

; 32 

33 

34 


1560 

1620 

1680 

1740 

1800 

1860 

1920 

1980 

2040 


1560 

1620 

1680 

1740 

1800 1860 1920 

1980 

2040 

0 

3632 

3468 

3310 

3158 

3010 

2868 

2730 

2596 

2467 

30 

3549 

3388 

3233 

3083 

2939 2798 2663 

2531 

2403 

1 

3629 

3465 

3307 

3155 

3008 

2866 

2728 

2594 

2465 

31 

3546 

3386 

3231 

3081 

2936 2796 

'2660 

2529 

2401 

2 

3626 

3463 

3305 

3153 

3005 

2863 

2725 

2592 

2462 

32 

3544 

3383 

3228 

3078 

2934 

12794 

2658 

2527 

2399 

3 

3623 

3460 

3302 

3150 

3003 

2861 

2723 

2590 

2460 

33 

3541 

3380 

3225 

3076 

2931 

|2792 

'2656 

2525 2397 

4 

3621 

3457 

3300 

3148 

3001 

2859 

2721 

2588 

2458 

34 

3538 

3378 

3223 

3073 

2929 

2789 

2654 

2522 2395 

5 

3618 

3454 

3297 

3145 

2998 

2856 

2719 

2585 

2456 

35 

3535 

3375 

3220 

3071 

2927 

2787 

2652 

2520,2393 

6 

3615 

3452 

3294 

3143 

2996 

2854 

2716 

2583 

2454 

36 

3533 

3372 

3218 

3069 

2924 

2785 

2649 

2518 2391 

73612 

3449 

3292 

3140 

2993 

2852 

2714 

2581 

2452 

37 

3530 

3370 

3215 

3066 

2922 

2782 

2647 

25162389 

83610 

3446 

3289 

3133 

2991 

2849 

2712 

2579 

2450 

38 

3527 

3367 

3213 

3064 

2920 

2780 

2645 

2514)2387 

93607 

3444 

3287 

3135 

2989 

2847 

2710 

2577 

2448 

39 

3525 

3365 

3210 

3061 

2917 

2778 

2643 

2512 2384 

10 3604 

3441 

3284 

3133 

2986 

2845 

2707 

2574 

2445 

40 

3522 

3362 

3208 

3059 

2915 

2775 

2640 

2510:2382 

11,3601 

3438 

3282 

3130 

2984 

2842 

2705 

2572 

2443 

41 

3519 

3359 

3205 

3056 

2912 

2773 

2638 

2507 

2380 

123598 

3436 

3279 

3128 

2981 

2840 

2703 

2570 

2441 

|42 

3516 

3357 

3203 

3054 

2910 

2771 

2636 

2505 

2378 

13 3596 

3433 

3276 

3125 

2979 

2838 

2701 2568 

2439 

143 

3514 

3354 

3200 

3052 

2903 

2769 

2634 

2503 2376 

143593 

3431 

3274 

3123 

2977 

2335 

2698 2566 

2437 

I 44 

3511 

3351 

3198 

3049 

2905 

2796 

2632 

2501 2374 

15 

3590 

3428 

3271 

3120 

2974 

2833 

2696 2564 

2435 

45 

3508 

3349 

3195 

3047 

2993 

2764 

2629 

249912372 

16 3587 

342513269 

3118 

2972 

2831 

2694 2561 

2433 

46 

3506 

3356 

3193 

3044 

2901 

2762 

2627 

2497 2370 

173585 

3423 

3266 

3115 

2969 

2828 

2692 2559 

2431 

47 

3503 

3344 

3190 

3042 

2898 

2760 

2625 

2494 2368 

1813582 

3420 

3264 

3113 

2967 

2826 

2689 2557 

2429 

48 

3500 

3341 

3188 

3039 

2896 

2757 

2623 

249212366 

193579 

3417 3261 

3110 

2965 

2824 

2687 2555 

2426 

49 

3497 

3338 

3185 

3337 

2894 

2755 

2621 

2490|2364 

203576 

3415 

3299 

3108 

2962 

2821 

2685 2553 

2424 

50 

3495 

3336 

3183 

3034 

2891 

2753 

2618 

2488; 

2362 

2l|3574 

34123256 

3105 

2960 

2819 

2683 2551 

2422 

51 

3492 

3333 

3180 

3032 

2889 

2750 

2616 

2486 

2359 

22 3571 

3409 32533103 

2958 

2317 

2681,2548 

2420 

52 

3489 

3331 

A 78 

3030 

2887 

2748 

2611 

2484! 

2357 

233568 

34073251 3101 

2955 

2815 

2678 2546 

2418 

53 

3487 

3328 

3175 

3027 

2884 

2746! 

2612 

2482 

2355 

243565 

3404 3248 

3098 

2953 

2812 

2675 2544 

2416 

54 

3484 

3325 

3173 

3025 

2882 

2744 

2610 

2480' 

2353 

25 

3563 

3401 3246 

3096 

2950 

2810 

2674 

2542 

2414 

55 

3481 

3323 

>170 

3022 

2880 

2741 

2607 

2477 2351 

26 

3560 

93993243 

3093 

2948 

2308 

2672 2540 

2412 

56 

3479 

3320 

3168 

3020 

2877 

2739 2605:2475 2349 

27 

3557 

3396,3241 

3091 

2946 

2805 2669 

2538 

2410 

57 

3476 

3318:3165 

3018 

2875 

2737 2603 2473 2347 

28 

3555 

3393 3238 

3088 

2943 

2803 

2667 

2535 

2408 

58 

3473 

33153163 

3015 

2873' 

2735 

260112471 2345 

29j3552 

3391 3236 

3086 

2941 

2801 

2665 

2533 

2405 

59 

3471 

3313 3160 

3013 

2870 

2732 2599.2469 2343 

30(3549 

3388,3233 

3083 

2939 

2798 

2663 

2531 

2403! 

6013468 

3310)3158 

3010 

286812730 2596 2467 2341 






























































































LUNAR TABLES 


291 


TABLE of Logistical Logarithms . 



35 

36 

37 

38 

39 

40 

41 

42 

43 


35 

36 

37 

38 

39 

40 

41 

42 

34 


2100 

2160 

o 

CO 

, 

CM 

f o 

CM 

1 CM 

1 CM 

2340 

2400 2460 

2520 

2580 


2100 

2160 

2220 

2280 

2340 

2400 

2460 

2520 

2580 

0 

2341 

2218 

2099 1984 

1871 

1761 

1654 

1549 

1447 

30 

2279 

2159 

2041 

1927 

1816 

1707 

1601 

1498 

1397 

1 

2339 

2216 

2098 1982 

1869 

1759 

1652 

1547 

1445 

31 

2277 

2157 

2039 

1925 

1814 

1705 

1599 

1496 

1395 

o 

2337 

2214 

2096 1980 1867 

1757 

1650 

1546 

1443 

32 

2275 

2155 

2037 

1923 

1812 

1703 

1598 

1494 

1393 

3 

2335 

2212 

2094 1978 

18651755 

1648 

1544 

1442 

33 

2273 

2153 

2035 

1921 

1810 

1702 

1596 

1493 

1392 

4 

2333 

2210 

2092 1976 

1863 

1754 

1647 

1542 

1440 

34 

2271 

2151 

2033 

1919 

1808 

1700 

1594 

1491 

1390 

5 

2331 

2208 

2090 

1974 

1862 

1752 

1645 

1540 

1438 

35 

2269 

2149 

2032 

1918 

1806 

1698 

1592 

1489 

1388 

6 

2328 

2206 

2088 

1972 

1860 

1750 

1643 

1539 

1437 

36 

2267 

2147 

2030 

1916 

1805 

1696 

1591 

1487 

1387 

7 

2326 

2204 

2086 

1970 

1858 

1748 

1641 

1537 

1435! 

37 

2265 

2145 

2028 

1914 

1803 

1694 

1589 

1486 

1385 

8 

2324 

2202 

2084 

1968 

1856 

1746 

1640 

1535 

1433 

38 

2263 

2143 

2026 

1912 

1801 

1693 

1587 

1484 

1383 

9 

2322 

2200 

2082 

1967 

1854 

1745 

1638 

1534 

1432 

39 

2261 

2141 

2024 

1910 

1799 

1691 

1585 

1482 

1382 

10 

2320 

2198 

2080 

1965 

1852 

1743 

1636 

1532 

1430 

40 

2259 

2139 

2022 

1908 

1797 

1689 

1584 

1481 

1380 

11 

2318 

2196 

2078 

1963 

1850 

1741 

1634 

1530 

1428 

41 

2257 

2137 

2020 

1906 

1795 

1687 

1582 

1479 

1378 

12 

2316 

2194 

2076 

1961 

1849 

1739 

1633 

1528 

1427 

42 

2255 

2135 

2018 

1904 

1794 

1686 

1580 

1477 

1377 

13 

2314 

2192 

2074 

1959 

1847 

1737 

1631 

1527 

1425 

43 

2253 

2133 

2016 

1903 

1792 

1684 1578 

1476 1375 

14 

2312 

2190 

2072 

1957 

1845 

1736 

1629 

1525 

1423 

44 

2251 

2131 

2014 

1901 

1790 

16821577 

1474 1373 

15 

2310 

2188 

2070 

1955 

1843 

1734 

1627 

1523 

1422 

45 

2249 

2129 

2012 

1899 

1788 

1680 1575 

1472 1372 

16 

2308 

2186 

2068 

1953 

1841 

1732 

1626 

1522 

1420 

46 

2247 

2127 

2010 

1897 

1786 

1678 1573 

1470 

1370 

17 

2306 

2184 

2066 

1951 1839 

1730 

1624 

1520 

14 i 8 

47 

2245 

2125 

2009 

1895 

1785 

16771571 

1469 

1368 

18 2304 

2182 

2064 

19501838 

1728 

1622 

1518 

1417 

48 

2243 

2123 

2007 

1893 

1783 

1675 1570 

1467 

1367 

19j 2302 

2180 

2062 

1948 1836 

1727 

1620 

1516 

1415 

49 

2241 

2121 

2005 

1891 

1781 

1673 1568 

1465 

1365 

20i2300 

2178 

2061 

1946 1834 

1725 

1619 

1515 

1413; 

50 

2239 

2119 

2003 

1889 

1779 

1671 1566 

1464 

1363 

21 2298 

2176 

2059 

1944 1832 

1723 

1617 

1513 

1412 

51 

2237 

2117 

2001 

1888 

1777 

1670 1565 

1462 

1362 

22 2296 

2174 

2057 

1942 1830 

1721 

1615 

1511 

1410 

52 

2235 

2115 

1999 

1886 

1775 

1668 1563 

1460 

1360 

23 2294 

2172 

2055 

1940 1828 

1719 

1613 

1510 

1408 

53 

2233 

2113 

1997 

1884 

1774 

1666 1561 

1459 

1359 

242291 

2170 

2053 

19381827 

1718 

1612 

1508 

1407 

54 

2231 

2111 

1995 

1882 

1772 

1664 1559 

1457 

1357 

25 2289 

2169 

2051 

1936 1825 

1716 

1610 

1506 

1405 

55 

2229 

2109 

1993 

1880 

1770 

1663 1558 

1455 

1355 

26 2287 

2167 

2049 

19341823 

1714 

1608 

1504 

1403 

56 

2227 

2107 

1991 

1878 

1768 

1661 

1556 

1451 

1354 

27 

;2285 

2165 

2047 

1933 1821 

1712 

1606 

1503 

1402 

57 

2225 

2105 

1989 

1876 

1766 

1659 1554 

1452 

1352 

28i2283 

2163 

2045 

1931 1819 

1711 

1605 

1501 

1400 

58 

2223 

2103 

1987 

1875 

1765 

1657 

1552 

1450 

1350 

29 2281 

2161 

204311929 1817 

1709 

1603 

1499 

1398 

59 

2220 

2101 

1986 

1873 

1763 

1655 1551 

1419 

1349 

302279 

2159 

204111927 1816 

1707 

1601 

1498 

1397 

60 

2218(2099 

1984 

• 

1871 

1761 

1654 1519 

1447 

1317 










































































392 


APPENDIX TO THE ASTRONOMY, 


TABLE of Logistical Logarithms. 



44 

45 

46 

1 47 

48 

49 

50 

51 

52 


44 

45 

46 

47 

48 

49 

I 50 

51 

52 


2640 

2700 

2760 

o 

CO 

©I 

2880 

2940 

3000 

3060 

3120 


2641 

12700 

2760 

2820 

288C 

>2940 

>3000 

| 

3060 

3120 

0 

1347 

1249 

1154 

1061 

969 

880 

792 

706 

621 

30 

1298 

1201 

1107 

1015 

924 

835 

74S 

663 

580 

1 

1345 

1248 

1152 

1059 

968 

878 

790 

704 

620 

31 

1296 

1200 

1105 

1013 

923 

834 

747 

662 

579 

2 

1344 

1246 

1151 

1057 

966 

877 

789 

703 

619 

32 

1295 

1198 

1104 

1012 

921 

833 

746 

661 

577 

o 

O 

1342 

1245 

1149 

1056 

963 

875 

787 

702 

617 

33 

1293 

1197 

1102 

1010 

920 

831 

744 

659 

576 

4 

1340 

1243 

1148 

1054 

963 

874 

786 

700 

616 

34 

1291 

1195 

1101 

1008 

918 

830 

743 

658 

574 

5 

1339 

1241 

1146 

1053 

96' 

872 

785 

699 

615 

35 

1290 

1193 

1099 

1007 

917 

828 

741 

656 

573 

6 

1337 

1240 

1145 

1051 

960 

871 

783 

697 

613 

36 

1288 

1192 

1098 

1005 

915 

827 

740 

655 

572 

7 

1335 

1238 

1143 

1050 

959 

869 

782 

696 

612 

37 

1287 

1190 

1096 

1004 

914 

825 

739 

654 

570 

8 

1334 

1237 

1141 

1048 

957 

868 

780 

694 

610 

38 

1285 

1189 

1095 

1002 

912 

824 

737 

652 

569 

9 

1332 

1235 

1140 

1047 

956 

866 

779 

693 

609 

39 

1283 

1187 

1093 

1001 

911 

822 

736 

651 

568 

10 

1331 

1233 

1138 

1045 

954 

865 

777 

692 

608 

40 

1282 

1186 

1091 

999 

909 

821 

734 

649 

566 

11 

1329 

1232 

1137 

1044 

953 

863 

776 

690 

606 

41 

1280 

1184 

1090 

998 

908 

819 

733 

648 

565 

12 

1327 

1230 

1135 

1042 

951 

862 

774 

689 

605 

42 

1278 

1182 

1088 

996 

906 

818 

731 

647 

563 

13 

1326 

1229 

1134 

1041 

! 950 

860 

773 

987 

603 

43 

1277 

1181 

1087 

995 

905 

816 

730 

645 

562 

14 

1324 

1227 

1132 

1039 

948 

859 

772 

686 

602 

44 

1275 

1179 

1085 

993 

903 

815 

729 

644 

561 

15 

1322 

1225 

1130 

1037 

947 

857 

770 

685 

601 

45 

1274 

1178 

1084 

992 

902 

814 

727 

642 

559 

18 

1321 1224 

1129 1036 

945 

856 

769 

683 

599 

46 

1272 

1176 

1082 

990 

900 

812 

726 

641 

558 

17 

131911222' 

11271034 

944 

855 

767 

682 

598 

47 

1270 

1174 

1081 

989 

899 

811 

724 

640 

557 

18 

1317 

1221 

1126 1033 

942 

853 

766 

680 

596 

48 

1269 

1173 

1079 

987 

897 

809 

723 

638 

555 

19 

1316 

1219 1124 1031 

941 

852 

764 

679 

595 

49 

1267 

1171 

1078 

986 

896 

808 

721 

637 

554 

20 

1314 

1217)1123 1030 

939 

850 

763 

678 

594 

50 

1266 

1170 

1076 

984 

894 

806 

720 

635 

552 

21 

1313 

1216)1121 1028 

938 

849 

762 

676 

592 

51 

1264 

1168 

1074 

983 

893 

805 

719 

634 

551 

22 

1311 

1214 1119 1027 

936 

847 

760 

675 

591 

52 

1262 

1167 

1073 

981 

891 

803 

717 

633 

550 

23 

130911213 1118 1025 

935 

846 

759 

673 

590 

53 

1261 

1165 

1071 

980 

890 

802 

716 

631 

548 

24 

1308| 1211 

1116 1024 

933 

844 

757 

672 

588 

54 1259, 

1163 

1070 

978 

888 

801 

714 

630 

547 

25 

1306 1209 

1115 

1022 

932 

843 

756 

670 

587 

55 1257 

1162 

1068 

977 

887 

799 

713 

628 

546 

26 

130411208 

1113 

1021 

930 

841 

754 

669 

585| 

56)1256 

1160 

1067 

975 

885 

798 

711 

627 

544 

27 

1303 

1206)1112 

1019 

929 

840 

753 

668 

584| 

571254 

1159 

1065 

974 

884 

796! 

710 

626 

543 

28 

1301 

1205 1110) 

1018 

927 

838 

751 

666 

583 ! 

58 1253 

1157 

1064 

972 

883 

795 

709 

624 

541 

29 

1300 

1203 1109 

1016 

926 

837 

750 

665 

5811 

59 1251 

1156 

1062 

971 

881 

793 

707 

623 

540 

30 

1298 

1201) 

1107(1015 

924 

835 

749 

663 

580jJ 

SOI 1249 

1154 

1061 

969 

880 

792) 

706 

621 

539 


% 






















































































LUNAR TABLES 


297 


TABLE of Logistical Logarithms. 



53 

j 54 

55 

i 56 

57 

58 

59 


53 

54 

55 

56 

57 

58 

59 


3180 

3240 

3300 

3360 

3420 

3480 

3540 


3180 

3240 

3300 

3360 

3420 

3480 

3540 

0 

539 

458 

378 

300 

223 

147 

73 

30 

498 

418 

339 

261 

185 

110 

36 

1 

537 

456 

377 

298 

221 

146 

72 

31 

497 

416 

337 

260 

184 

109 

35 

2 

536 

455 

375 

297 

220 

145 

71 

32 

495 

415 

336 

258 

182 

107 

34 

3 

535 

454 

374 

296 

219 

143 

69 

33 

494 

414 

335 

257 

181 

106 

33 

4 

533 

452 

373 

294 

218 

142 

68 

34 

493 

412 

333 

256 

180 

105 

31 

5 

532 

451 

371 

293 

216 

141 

67 

35 

491 

411 

332 

255 

179 

104 

30 

6 

531 

450 

370 

292 

215 

140 

66 

36 

490 

410 

331 

253 

177 

103 

29 

7 

529 

448 

369 

291 

214 

139 

64 

37 

489 

408 

329 

252 

176 

101 

28 

8 

528 

447 

367 

289 

213 

137 

63 | 

38 

487 

407 

328 

251 

175 

100 

27 

9 

526 

446 

366 

288 

211 

136 

62 ! 

39 

486 

406 

327 

250 

174 

99 

25 

10 

525 

444 

365 

287 

210 

135 

61 

40 

484 

404 

326 

248 

172 

98 

24 

11 

524 

443 

363 

285 

209 

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